Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series Appell series and beyond
Michael J. Schlosser Faculty of Mathematics
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Outline
1 Appell hypergeometric series
2 Contiguous relations
3 Partial differential equations
4 Integral representations
5 Transformations
6 Reduction formulae
7 Extensions
8 A curious integral
9 More sums
Multiple hypergeometric series – Appell series and beyond Recall the Pochhammer symbol notation for the shifted factorial: ( a(a + 1) ... (a + n − 1) if n = 1, 2,... , (a)n := 1 if n = 0.
The (generalized) hypergeometric series is defined by a1, a2,..., ar X (a1)n (a2)n ... (ar )n n r Fs ; x = x . b1, b2,..., bs n!(b1)n ... (bs )n n≥0
Goal: We would like to generalize the Gauß hypergeometric function a, b X (a)n (b)n n 2F1 ; x = x c n!(c)n n≥0
to a double series depending on two variables.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Appell hypergeometric series
Multiple hypergeometric series – Appell series and beyond Goal: We would like to generalize the Gauß hypergeometric function a, b X (a)n (b)n n 2F1 ; x = x c n!(c)n n≥0
to a double series depending on two variables.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Appell hypergeometric series
Recall the Pochhammer symbol notation for the shifted factorial: ( a(a + 1) ... (a + n − 1) if n = 1, 2,... , (a)n := 1 if n = 0.
The (generalized) hypergeometric series is defined by a1, a2,..., ar X (a1)n (a2)n ... (ar )n n r Fs ; x = x . b1, b2,..., bs n!(b1)n ... (bs )n n≥0
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Appell hypergeometric series
Recall the Pochhammer symbol notation for the shifted factorial: ( a(a + 1) ... (a + n − 1) if n = 1, 2,... , (a)n := 1 if n = 0.
The (generalized) hypergeometric series is defined by a1, a2,..., ar X (a1)n (a2)n ... (ar )n n r Fs ; x = x . b1, b2,..., bs n!(b1)n ... (bs )n n≥0
Goal: We would like to generalize the Gauß hypergeometric function a, b X (a)n (b)n n 2F1 ; x = x c n!(c)n n≥0
to a double series depending on two variables.
Multiple hypergeometric series – Appell series and beyond We consider the product
0 0 0 0 a, b a , b X X (a)m (a )n (b)m (b )n m n 2F1 ; x 2F1 0 ; y = 0 x y . c c m! n!(c)m (c )n m≥0 n≥0
0 0 Now replace one, two or three of the products (a)m (a )n,(b)m (b )n, 0 (c)m (c )n by the corresponding expressions
(a)m+n, (b)m+n, (c)m+n.
There are five possibilities, one of which gives the series X X (a)m+n (b)m+m m n a, b x y = 2F1 ; x + y . m! n!(c)m+n c m≥0 n≥0
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In the following, we follow, to great extent, the classical expositions from W.N. Bailey, Generalized hypergeometric series, CUP, 1935, and L.J. Slater, Generalized hypergeometric functions, CUP, 1966.
Multiple hypergeometric series – Appell series and beyond 0 0 Now replace one, two or three of the products (a)m (a )n,(b)m (b )n, 0 (c)m (c )n by the corresponding expressions
(a)m+n, (b)m+n, (c)m+n.
There are five possibilities, one of which gives the series X X (a)m+n (b)m+m m n a, b x y = 2F1 ; x + y . m! n!(c)m+n c m≥0 n≥0
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In the following, we follow, to great extent, the classical expositions from W.N. Bailey, Generalized hypergeometric series, CUP, 1935, and L.J. Slater, Generalized hypergeometric functions, CUP, 1966. We consider the product
0 0 0 0 a, b a , b X X (a)m (a )n (b)m (b )n m n 2F1 ; x 2F1 0 ; y = 0 x y . c c m! n!(c)m (c )n m≥0 n≥0
Multiple hypergeometric series – Appell series and beyond There are five possibilities, one of which gives the series X X (a)m+n (b)m+m m n a, b x y = 2F1 ; x + y . m! n!(c)m+n c m≥0 n≥0
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In the following, we follow, to great extent, the classical expositions from W.N. Bailey, Generalized hypergeometric series, CUP, 1935, and L.J. Slater, Generalized hypergeometric functions, CUP, 1966. We consider the product
0 0 0 0 a, b a , b X X (a)m (a )n (b)m (b )n m n 2F1 ; x 2F1 0 ; y = 0 x y . c c m! n!(c)m (c )n m≥0 n≥0
0 0 Now replace one, two or three of the products (a)m (a )n,(b)m (b )n, 0 (c)m (c )n by the corresponding expressions
(a)m+n, (b)m+n, (c)m+n.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In the following, we follow, to great extent, the classical expositions from W.N. Bailey, Generalized hypergeometric series, CUP, 1935, and L.J. Slater, Generalized hypergeometric functions, CUP, 1966. We consider the product
0 0 0 0 a, b a , b X X (a)m (a )n (b)m (b )n m n 2F1 ; x 2F1 0 ; y = 0 x y . c c m! n!(c)m (c )n m≥0 n≥0
0 0 Now replace one, two or three of the products (a)m (a )n,(b)m (b )n, 0 (c)m (c )n by the corresponding expressions
(a)m+n, (b)m+n, (c)m+n.
There are five possibilities, one of which gives the series X X (a)m+n (b)m+m m n a, b x y = 2F1 ; x + y . m! n!(c)m+n c m≥0 n≥0
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Four remaining possibilities (Paul Appell [1855–1930], 1880; and P. Appell & Marie-Joseph Kamp´ede F´eriet [1893–1982], 1926):
0 0 X X (a)m+n (b)m (b )n m n F1 a; b, b ; c; x, y := x y , |x|, |y| < 1. m! n!(c)m+n m≥0 n≥0
0 X X (a)m+n (b)m (b )n 0 0 m n |x| + |y| < 1 F2 a; b, b ; c, c ; x, y := 0 x y , . m! n!(c)m (c )n m≥0 n≥0
0 0 0 0 X X (a)m (a )n (b)m (b )n m n F3 a, a ; b, b ; c; x, y := x y , |x|, |y| < 1. m! n!(c)m+n m≥0 n≥0
0 X X (a)m+n (b)m+n m n 1 1 |x| 2 + |y| 2 < 1 F4 a; b; c, c ; x, y := 0 x y , . m! n!(c)m (c )n m≥0 n≥0
Multiple hypergeometric series – Appell series and beyond 0 0 0 0 0 F1 a; b, b ; c; x, 0 = F2 a; b, b ; c, c ; x, 0 = F3 a, a ; b, b ; c; x, 0 a, b = F a; b; c, c0; x, 0 = F ; x . 4 2 1 c
0 F1 a; b, 0; c; x, y = F2 a; b, 0; c, c ; x, y a, b = F a, a0; b, 0; c; x, y = F ; x . 3 2 1 c
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Simple observations:
0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y . m!(c)m c + m m≥0
Multiple hypergeometric series – Appell series and beyond 0 F1 a; b, 0; c; x, y = F2 a; b, 0; c, c ; x, y a, b = F a, a0; b, 0; c; x, y = F ; x . 3 2 1 c
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Simple observations:
0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y . m!(c)m c + m m≥0
0 0 0 0 0 F1 a; b, b ; c; x, 0 = F2 a; b, b ; c, c ; x, 0 = F3 a, a ; b, b ; c; x, 0 a, b = F a; b; c, c0; x, 0 = F ; x . 4 2 1 c
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Simple observations:
0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y . m!(c)m c + m m≥0
0 0 0 0 0 F1 a; b, b ; c; x, 0 = F2 a; b, b ; c, c ; x, 0 = F3 a, a ; b, b ; c; x, 0 a, b = F a; b; c, c0; x, 0 = F ; x . 4 2 1 c
0 F1 a; b, 0; c; x, y = F2 a; b, 0; c, c ; x, y a, b = F a, a0; b, 0; c; x, y = F ; x . 3 2 1 c
Multiple hypergeometric series – Appell series and beyond All contiguous relations for the F1 can be derived from these four:
0 0 0 (a − b − b ) F1 a; b, b ; c; x, y − aF 1 a + 1; b, b ; c; x, y 0 0 0 +bF 1 a; b + 1, b ; c; x, y + b F1 a; b, b + 1; c; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y − (c − a) F1 a; b, b ; c + 1; x, y 0 −aF 1 a + 1; b, b ; c + 1; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y + c(x − 1) F1 a; b + 1, b ; c; x, y 0 −(c − a)xF 1 a; b + 1, b ; c + 1; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y + c(y − 1) F1 a; b, b + 1; c; x, y 0 −(c − a)yF 1 a; b, b + 1; c + 1; x, y = 0.
Similar sets of relations exist for the other Appell functions. See R.G. Buschman, Contiguous relations for Appell functions, J. Indian Math. Soc. 29 (1987), 165–171.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Contiguous relations
Multiple hypergeometric series – Appell series and beyond Similar sets of relations exist for the other Appell functions. See R.G. Buschman, Contiguous relations for Appell functions, J. Indian Math. Soc. 29 (1987), 165–171.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Contiguous relations
All contiguous relations for the F1 can be derived from these four:
0 0 0 (a − b − b ) F1 a; b, b ; c; x, y − aF 1 a + 1; b, b ; c; x, y 0 0 0 +bF 1 a; b + 1, b ; c; x, y + b F1 a; b, b + 1; c; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y − (c − a) F1 a; b, b ; c + 1; x, y 0 −aF 1 a + 1; b, b ; c + 1; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y + c(x − 1) F1 a; b + 1, b ; c; x, y 0 −(c − a)xF 1 a; b + 1, b ; c + 1; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y + c(y − 1) F1 a; b, b + 1; c; x, y 0 −(c − a)yF 1 a; b, b + 1; c + 1; x, y = 0.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Contiguous relations
All contiguous relations for the F1 can be derived from these four:
0 0 0 (a − b − b ) F1 a; b, b ; c; x, y − aF 1 a + 1; b, b ; c; x, y 0 0 0 +bF 1 a; b + 1, b ; c; x, y + b F1 a; b, b + 1; c; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y − (c − a) F1 a; b, b ; c + 1; x, y 0 −aF 1 a + 1; b, b ; c + 1; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y + c(x − 1) F1 a; b + 1, b ; c; x, y 0 −(c − a)xF 1 a; b + 1, b ; c + 1; x, y = 0,
0 0 cF 1 a; b, b ; c; x, y + c(y − 1) F1 a; b, b + 1; c; x, y 0 −(c − a)yF 1 a; b, b + 1; c + 1; x, y = 0.
Similar sets of relations exist for the other Appell functions. See R.G. Buschman, Contiguous relations for Appell functions, J. Indian Math. Soc. 29 (1987), 165–171.
Multiple hypergeometric series – Appell series and beyond Let 0 X X m n z = F1 a; b, b ; c; x, y = Am,nx y . m≥0 n≥0 Then (a + m + n)(b + m) A = A , m+1,n (1 + m)(c + m + n) m,n and (a + m + n)(b0 + n) A = A . m,n+1 (1 + n)(c + m + n) m,n Denoting ∂ ∂ θ = x and φ = y , ∂x ∂y
we see that F1 satisfies the partial differential equations 1 (θ + φ + a)(θ + b) − θ(θ + φ + c − 1)z = 0, x 1 (θ + φ + a)(φ + b0) − φ(θ + φ + c − 1)z = 0. x
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Partial differential equations
Multiple hypergeometric series – Appell series and beyond Denoting ∂ ∂ θ = x and φ = y , ∂x ∂y
we see that F1 satisfies the partial differential equations 1 (θ + φ + a)(θ + b) − θ(θ + φ + c − 1)z = 0, x 1 (θ + φ + a)(φ + b0) − φ(θ + φ + c − 1)z = 0. x
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Partial differential equations
Let 0 X X m n z = F1 a; b, b ; c; x, y = Am,nx y . m≥0 n≥0 Then (a + m + n)(b + m) A = A , m+1,n (1 + m)(c + m + n) m,n and (a + m + n)(b0 + n) A = A . m,n+1 (1 + n)(c + m + n) m,n
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Partial differential equations
Let 0 X X m n z = F1 a; b, b ; c; x, y = Am,nx y . m≥0 n≥0 Then (a + m + n)(b + m) A = A , m+1,n (1 + m)(c + m + n) m,n and (a + m + n)(b0 + n) A = A . m,n+1 (1 + n)(c + m + n) m,n Denoting ∂ ∂ θ = x and φ = y , ∂x ∂y
we see that F1 satisfies the partial differential equations 1 (θ + φ + a)(θ + b) − θ(θ + φ + c − 1)z = 0, x 1 (θ + φ + a)(φ + b0) − φ(θ + φ + c − 1)z = 0. x
Multiple hypergeometric series – Appell series and beyond Then z = F1 satisfies the partial differential equations
x(1 − x)r + y(1 − x)s + [c − (a + b + 1)x]p − byq − abz = 0, y(1 − y)t + x(1 − y)s + [c − (a + b0 + 1)y]q − b0xp − ab0z = 0.
Similarly, z = F2 satisfies the partial differential equations
x(1 − x)r − xys + [c − (a + b + 1)x]p − byq − abz = 0, y(1 − y)t − xys + [c0 − (a + b0 + 1)y]q − b0xp − ab0z = 0.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Now let ∂z ∂z ∂z ∂z ∂z2 ∂z2 p = , q = , r = , s = , t = . ∂x ∂y ∂x ∂y ∂x 2 ∂y 2
Multiple hypergeometric series – Appell series and beyond Similarly, z = F2 satisfies the partial differential equations
x(1 − x)r − xys + [c − (a + b + 1)x]p − byq − abz = 0, y(1 − y)t − xys + [c0 − (a + b0 + 1)y]q − b0xp − ab0z = 0.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Now let ∂z ∂z ∂z ∂z ∂z2 ∂z2 p = , q = , r = , s = , t = . ∂x ∂y ∂x ∂y ∂x 2 ∂y 2
Then z = F1 satisfies the partial differential equations
x(1 − x)r + y(1 − x)s + [c − (a + b + 1)x]p − byq − abz = 0, y(1 − y)t + x(1 − y)s + [c − (a + b0 + 1)y]q − b0xp − ab0z = 0.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Now let ∂z ∂z ∂z ∂z ∂z2 ∂z2 p = , q = , r = , s = , t = . ∂x ∂y ∂x ∂y ∂x 2 ∂y 2
Then z = F1 satisfies the partial differential equations
x(1 − x)r + y(1 − x)s + [c − (a + b + 1)x]p − byq − abz = 0, y(1 − y)t + x(1 − y)s + [c − (a + b0 + 1)y]q − b0xp − ab0z = 0.
Similarly, z = F2 satisfies the partial differential equations
x(1 − x)r − xys + [c − (a + b + 1)x]p − byq − abz = 0, y(1 − y)t − xys + [c0 − (a + b0 + 1)y]q − b0xp − ab0z = 0.
Multiple hypergeometric series – Appell series and beyond and z = F4 satisfies the partial differential equations
x(1 − x)r − y 2t − 2xys + cp − (a + b + 1)(xp + yq) − abz = 0, y(1 − y)t − x 2r − 2xys + c0q − (a + b + 1)(xp + yq) − abz = 0.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Similarly, z = F3 satisfies the partial differential equations
x(1 − x)r + ys + [c − (a + b + 1)x]p − abz = 0, y(1 − y)t + xs + [c − (a0 + b0 + 1)y]q − a0b0z = 0.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Similarly, z = F3 satisfies the partial differential equations
x(1 − x)r + ys + [c − (a + b + 1)x]p − abz = 0, y(1 − y)t + xs + [c − (a0 + b0 + 1)y]q − a0b0z = 0.
and z = F4 satisfies the partial differential equations
x(1 − x)r − y 2t − 2xys + cp − (a + b + 1)(xp + yq) − abz = 0, y(1 − y)t − x 2r − 2xys + c0q − (a + b + 1)(xp + yq) − abz = 0.
Multiple hypergeometric series – Appell series and beyond Consider the integral
ZZ 0 0 I = ub−1v b −1(1 − u − v)c−b−b −1(1 − ux − vy)−a du dv,
taken over the triangular region u ≥ 0, v ≥ 0, u + v ≤ 1. (We also assume suitable conditions of the parameters a, b, b0, c such that the integral converges.) Now, provided |vy/(1 − ux)| < 1, we have, by binomial expansion, m X (a)m vy (1 − ux − vy)−a = (1 − ux)−a (1)m 1 − ux m≥0
X (a)m = v my m(1 − ux)−a−m (1)m m≥0
X (a)m X (a + m)n = v my m unx n. (1)m (1)n m≥0 n≥0
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Integral representations
Multiple hypergeometric series – Appell series and beyond Now, provided |vy/(1 − ux)| < 1, we have, by binomial expansion, m X (a)m vy (1 − ux − vy)−a = (1 − ux)−a (1)m 1 − ux m≥0
X (a)m = v my m(1 − ux)−a−m (1)m m≥0
X (a)m X (a + m)n = v my m unx n. (1)m (1)n m≥0 n≥0
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Integral representations
Consider the integral
ZZ 0 0 I = ub−1v b −1(1 − u − v)c−b−b −1(1 − ux − vy)−a du dv,
taken over the triangular region u ≥ 0, v ≥ 0, u + v ≤ 1. (We also assume suitable conditions of the parameters a, b, b0, c such that the integral converges.)
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Integral representations
Consider the integral
ZZ 0 0 I = ub−1v b −1(1 − u − v)c−b−b −1(1 − ux − vy)−a du dv,
taken over the triangular region u ≥ 0, v ≥ 0, u + v ≤ 1. (We also assume suitable conditions of the parameters a, b, b0, c such that the integral converges.) Now, provided |vy/(1 − ux)| < 1, we have, by binomial expansion, m X (a)m vy (1 − ux − vy)−a = (1 − ux)−a (1)m 1 − ux m≥0
X (a)m = v my m(1 − ux)−a−m (1)m m≥0
X (a)m X (a + m)n = v my m unx n. (1)m (1)n m≥0 n≥0
Multiple hypergeometric series – Appell series and beyond which yields
b, b0, c − b − b0 I = Γ F a; b, b0; c; x, y. c 1
(While I is a double integral, a single integral for F1 even exists. We will turn to that later.)
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Thus, ZZ X X (a)m+n 0 0 I = x ny m ub−1+nv b −1+m(1 − u − v)c−b−b −1 du dv (1)m(1)n m≥0 n≥0 0 0 X X (a)m+n b + n, b + m, c − b − b = x ny m Γ , (1)m(1)n c + m + n m≥0 n≥0
Multiple hypergeometric series – Appell series and beyond (While I is a double integral, a single integral for F1 even exists. We will turn to that later.)
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Thus, ZZ X X (a)m+n 0 0 I = x ny m ub−1+nv b −1+m(1 − u − v)c−b−b −1 du dv (1)m(1)n m≥0 n≥0 0 0 X X (a)m+n b + n, b + m, c − b − b = x ny m Γ , (1)m(1)n c + m + n m≥0 n≥0
which yields
b, b0, c − b − b0 I = Γ F a; b, b0; c; x, y. c 1
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Thus, ZZ X X (a)m+n 0 0 I = x ny m ub−1+nv b −1+m(1 − u − v)c−b−b −1 du dv (1)m(1)n m≥0 n≥0 0 0 X X (a)m+n b + n, b + m, c − b − b = x ny m Γ , (1)m(1)n c + m + n m≥0 n≥0
which yields
b, b0, c − b − b0 I = Γ F a; b, b0; c; x, y. c 1
(While I is a double integral, a single integral for F1 even exists. We will turn to that later.)
Multiple hypergeometric series – Appell series and beyond and
ZZ 0 0 0 ub−1v b −1(1 − u − v)c−b−b −1(1 − ux)−a(1 − vy)−a du dv
b, b0, c − b − b0 = Γ F a, a0; b, b0; c0; x, y, c0 3
the last integral taken over the triangular region u ≥ 0, v ≥ 0, u + v ≤ 1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Simlarly,
1 1 Z Z 0 0 0 0 ub−1v b −1(1 − u)c−b −1(1 − v)c −b −1(1 − ux − vy)−a du dv 0 0 b, b0, c − b, c0 − b0 = Γ F a; b, b0; c, c0; x, y, c, c0 2
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Simlarly,
1 1 Z Z 0 0 0 0 ub−1v b −1(1 − u)c−b −1(1 − v)c −b −1(1 − ux − vy)−a du dv 0 0 b, b0, c − b, c0 − b0 = Γ F a; b, b0; c, c0; x, y, c, c0 2
and
ZZ 0 0 0 ub−1v b −1(1 − u − v)c−b−b −1(1 − ux)−a(1 − vy)−a du dv
b, b0, c − b − b0 = Γ F a, a0; b, b0; c0; x, y, c0 3
the last integral taken over the triangular region u ≥ 0, v ≥ 0, u + v ≤ 1.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
The double integral for F4 is more complicated:
1 1 Z Z 0 ua−1v b−1(1 − u)c−a−1(1 − v)c −b−1(1 − ux)−b(1 − vy)−a 0 0 0 uvxy c+c −a−b−1 × 1 − du dv (1 − ux)(1 − vy) a, b, c − a, c0 − b = Γ F a; b; c, c0; x(1 − y), y(1 − x). c, c0 4
Multiple hypergeometric series – Appell series and beyond Let
1 Z 0 I 0 = ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, 0 where
Z 1 0 X X (b)m (b )n I 0 = ua−1(1 − u)c−a−1 umx m uny n du (1)m (1)n m≥0 n≥0 0 0 Z 1 X X (b)m(b )n = x my n ua+m+n−1(1 − u)c−a−1 du (1)m(1)n m≥0 n≥0 0 0 X X (b)m(b )n a + m + n, c − a = x my n Γ , (1)m(1)n c + m + n m≥0 n≥0
hence a, c − a I 0 = Γ F a; b, b0; c; x, y. c 1
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1881, Emile´ Picard discovered a single integral for F1.
Multiple hypergeometric series – Appell series and beyond Then
Z 1 0 X X (b)m (b )n I 0 = ua−1(1 − u)c−a−1 umx m uny n du (1)m (1)n m≥0 n≥0 0 0 Z 1 X X (b)m(b )n = x my n ua+m+n−1(1 − u)c−a−1 du (1)m(1)n m≥0 n≥0 0 0 X X (b)m(b )n a + m + n, c − a = x my n Γ , (1)m(1)n c + m + n m≥0 n≥0
hence a, c − a I 0 = Γ F a; b, b0; c; x, y. c 1
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1881, Emile´ Picard discovered a single integral for F1. Let
1 Z 0 I 0 = ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, 0 where
Multiple hypergeometric series – Appell series and beyond hence a, c − a I 0 = Γ F a; b, b0; c; x, y. c 1
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1881, Emile´ Picard discovered a single integral for F1. Let
1 Z 0 I 0 = ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, 0 where
Z 1 0 X X (b)m (b )n I 0 = ua−1(1 − u)c−a−1 umx m uny n du (1)m (1)n m≥0 n≥0 0 0 Z 1 X X (b)m(b )n = x my n ua+m+n−1(1 − u)c−a−1 du (1)m(1)n m≥0 n≥0 0 0 X X (b)m(b )n a + m + n, c − a = x my n Γ , (1)m(1)n c + m + n m≥0 n≥0
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1881, Emile´ Picard discovered a single integral for F1. Let
1 Z 0 I 0 = ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, 0 where
Z 1 0 X X (b)m (b )n I 0 = ua−1(1 − u)c−a−1 umx m uny n du (1)m (1)n m≥0 n≥0 0 0 Z 1 X X (b)m(b )n = x my n ua+m+n−1(1 − u)c−a−1 du (1)m(1)n m≥0 n≥0 0 0 X X (b)m(b )n a + m + n, c − a = x my n Γ , (1)m(1)n c + m + n m≥0 n≥0
hence a, c − a I 0 = Γ F a; b, b0; c; x, y. c 1
Multiple hypergeometric series – Appell series and beyond Z φ dθ F (φ, k) = p 2 0 1 − k2 sin θ 1 1 1 3 π = sin φ F ; , ; ; sin2 φ, k2 sin2 φ , |<φ| < , 1 2 2 2 2 2
Z φ p E(φ, k) = 1 − k2 sin2 θ dθ 0 1 1 1 3 π = sin φ F ; , − ; ; sin2 φ, k2 sin2 φ , |<φ| < , 1 2 2 2 2 2
Z π/2 dθ π 1 1 2 Π(n, k) = = F1 ; 1, ; 1; n, k . 2 p 2 0 (1 − n sin θ) 1 − k2 sin θ 2 2 2
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It follows that the incomplete elliptic integrals F and E and the complete elliptic integral Π can all be expressed in terms of special cases of the F1:
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
It follows that the incomplete elliptic integrals F and E and the complete elliptic integral Π can all be expressed in terms of special cases of the F1:
Z φ dθ F (φ, k) = p 2 0 1 − k2 sin θ 1 1 1 3 π = sin φ F ; , ; ; sin2 φ, k2 sin2 φ , |<φ| < , 1 2 2 2 2 2
Z φ p E(φ, k) = 1 − k2 sin2 θ dθ 0 1 1 1 3 π = sin φ F ; , − ; ; sin2 φ, k2 sin2 φ , |<φ| < , 1 2 2 2 2 2
Z π/2 dθ π 1 1 2 Π(n, k) = = F1 ; 1, ; 1; n, k . 2 p 2 0 (1 − n sin θ) 1 − k2 sin θ 2 2 2
Multiple hypergeometric series – Appell series and beyond In the single integral for the F1 series,
0 F1 a; b, b ; c; x, y 1 c Z 0 = Γ ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, a, c − a 0 one may use the substitution of variables
u = 1 − v
to prove 0 x y F a; b, b0; c; x, y = (1−x)−b(1−y)−b F c − a; b, b0; c; , . 1 1 x − 1 y − 1
0 For b = 0 this reduces to the Pfaff–Kummer transformation for the 2F1: a, b c − a, b x F ; x = (1 − x)−b F ; . 2 1 c 2 1 c x − 1
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Transformations
Multiple hypergeometric series – Appell series and beyond 0 For b = 0 this reduces to the Pfaff–Kummer transformation for the 2F1: a, b c − a, b x F ; x = (1 − x)−b F ; . 2 1 c 2 1 c x − 1
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Transformations
In the single integral for the F1 series,
0 F1 a; b, b ; c; x, y 1 c Z 0 = Γ ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, a, c − a 0 one may use the substitution of variables
u = 1 − v
to prove 0 x y F a; b, b0; c; x, y = (1−x)−b(1−y)−b F c − a; b, b0; c; , . 1 1 x − 1 y − 1
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Transformations
In the single integral for the F1 series,
0 F1 a; b, b ; c; x, y 1 c Z 0 = Γ ua−1(1 − u)c−a−1(1 − ux)−b(1 − uy)−b du, a, c − a 0 one may use the substitution of variables
u = 1 − v
to prove 0 x y F a; b, b0; c; x, y = (1−x)−b(1−y)−b F c − a; b, b0; c; , . 1 1 x − 1 y − 1
0 For b = 0 this reduces to the Pfaff–Kummer transformation for the 2F1: a, b c − a, b x F ; x = (1 − x)−b F ; . 2 1 c 2 1 c x − 1
Multiple hypergeometric series – Appell series and beyond For b0 = 0 this reduces again to the Pfaff–Kummer transformation for the 2F1 series. On the other hand, if c = b + b0, then a, b0 y − x F a; b, b0; b + b0; x, y = (1 − x)−a F ; 1 2 1 b + b0 1 − x a, b x − y = (1 − y)−a F ; . 2 1 b + b0 1 − y
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Similarly, the substitution of variables v u = 1 − x + vx can be used to prove
x y − x F a; b, b0; c; x, y = (1 − x)−aF a; −b − b0 + c, b0; c; , . 1 1 x − 1 1 − x
Multiple hypergeometric series – Appell series and beyond On the other hand, if c = b + b0, then a, b0 y − x F a; b, b0; b + b0; x, y = (1 − x)−a F ; 1 2 1 b + b0 1 − x a, b x − y = (1 − y)−a F ; . 2 1 b + b0 1 − y
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Similarly, the substitution of variables v u = 1 − x + vx can be used to prove
x y − x F a; b, b0; c; x, y = (1 − x)−aF a; −b − b0 + c, b0; c; , . 1 1 x − 1 1 − x
For b0 = 0 this reduces again to the Pfaff–Kummer transformation for the 2F1 series.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Similarly, the substitution of variables v u = 1 − x + vx can be used to prove
x y − x F a; b, b0; c; x, y = (1 − x)−aF a; −b − b0 + c, b0; c; , . 1 1 x − 1 1 − x
For b0 = 0 this reduces again to the Pfaff–Kummer transformation for the 2F1 series. On the other hand, if c = b + b0, then a, b0 y − x F a; b, b0; b + b0; x, y = (1 − x)−a F ; 1 2 1 b + b0 1 − x a, b x − y = (1 − y)−a F ; . 2 1 b + b0 1 − y
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Similarly,
x − y y F a; b, b0; c; x, y = (1 − y)−aF a; b, c − b − b0; c; , , 1 1 1 − y y − 1
0 F1 a; b, b ; c; x, y 0 x − y = (1 − x)c−a−b(1 − y)−b F c − a; c − b − b0, b0; c; x, , 1 1 − y
0 F1 a; b, b ; c; x, y 0 y − x = (1 − x)−b(1 − y)c−a−b F c − a; b, c − b − b0; c; , y . 1 1 − x
Multiple hypergeometric series – Appell series and beyond Also quadratic transformations are known for Appell functions. See B.C. Carlson, Quadratic transformations of Appell functions, SIAM J. Math. Anal. 7 (1976), 291–304.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Similarly,
x y F a; b, b0; c, c0; x, y = (1 − x)−aF a; c − b, b0; c, c0; , , 2 2 x − 1 1 − x
x y F a; b, b0; c, c0; x, y = (1 − y)−aF a; b, c0 − b0; c, c0; , , 2 2 1 − y y − 1
0 0 F2 a; b, b ; c, c ; x, y x y = (1 − x − y)−aF a; c − a, c0 − b0; c, c0; , . 2 x + y − 1 x + y − 1
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Similarly,
x y F a; b, b0; c, c0; x, y = (1 − x)−aF a; c − b, b0; c, c0; , , 2 2 x − 1 1 − x
x y F a; b, b0; c, c0; x, y = (1 − y)−aF a; b, c0 − b0; c, c0; , , 2 2 1 − y y − 1
0 0 F2 a; b, b ; c, c ; x, y x y = (1 − x − y)−aF a; c − a, c0 − b0; c, c0; , . 2 x + y − 1 x + y − 1
Also quadratic transformations are known for Appell functions. See B.C. Carlson, Quadratic transformations of Appell functions, SIAM J. Math. Anal. 7 (1976), 291–304.
Multiple hypergeometric series – Appell series and beyond The transformations imply the following reduction formulae:
F1[y = x]: 0 0 c − a, c − b − b F a; b, b0; c; x, x = (1 − x)c−a−b−b F ; x . 1 2 1 c By Euler’s transformation this is a, b + b0 F a; b, b0; c; x, x = F ; x . 1 2 1 c
0 F1[c = b + b ]: a, b x − y F a; b, b0; b + b0; x, y = (1 − y)−a F ; . 1 2 1 b + b0 1 − y
F2[c = b]: a, b0 y F a; b, b0; b, c0; x, y = (1 − x)−a F ; . 2 2 1 c0 1 − x
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Reduction formulae
Multiple hypergeometric series – Appell series and beyond F1[y = x]: 0 0 c − a, c − b − b F a; b, b0; c; x, x = (1 − x)c−a−b−b F ; x . 1 2 1 c By Euler’s transformation this is a, b + b0 F a; b, b0; c; x, x = F ; x . 1 2 1 c
0 F1[c = b + b ]: a, b x − y F a; b, b0; b + b0; x, y = (1 − y)−a F ; . 1 2 1 b + b0 1 − y
F2[c = b]: a, b0 y F a; b, b0; b, c0; x, y = (1 − x)−a F ; . 2 2 1 c0 1 − x
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Reduction formulae
The transformations imply the following reduction formulae:
Multiple hypergeometric series – Appell series and beyond 0 F1[c = b + b ]: a, b x − y F a; b, b0; b + b0; x, y = (1 − y)−a F ; . 1 2 1 b + b0 1 − y
F2[c = b]: a, b0 y F a; b, b0; b, c0; x, y = (1 − x)−a F ; . 2 2 1 c0 1 − x
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Reduction formulae
The transformations imply the following reduction formulae:
F1[y = x]: 0 0 c − a, c − b − b F a; b, b0; c; x, x = (1 − x)c−a−b−b F ; x . 1 2 1 c By Euler’s transformation this is a, b + b0 F a; b, b0; c; x, x = F ; x . 1 2 1 c
Multiple hypergeometric series – Appell series and beyond F2[c = b]: a, b0 y F a; b, b0; b, c0; x, y = (1 − x)−a F ; . 2 2 1 c0 1 − x
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Reduction formulae
The transformations imply the following reduction formulae:
F1[y = x]: 0 0 c − a, c − b − b F a; b, b0; c; x, x = (1 − x)c−a−b−b F ; x . 1 2 1 c By Euler’s transformation this is a, b + b0 F a; b, b0; c; x, x = F ; x . 1 2 1 c
0 F1[c = b + b ]: a, b x − y F a; b, b0; b + b0; x, y = (1 − y)−a F ; . 1 2 1 b + b0 1 − y
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Reduction formulae
The transformations imply the following reduction formulae:
F1[y = x]: 0 0 c − a, c − b − b F a; b, b0; c; x, x = (1 − x)c−a−b−b F ; x . 1 2 1 c By Euler’s transformation this is a, b + b0 F a; b, b0; c; x, x = F ; x . 1 2 1 c
0 F1[c = b + b ]: a, b x − y F a; b, b0; b + b0; x, y = (1 − y)−a F ; . 1 2 1 b + b0 1 − y
F2[c = b]: a, b0 y F a; b, b0; b, c0; x, y = (1 − x)−a F ; . 2 2 1 c0 1 − x
Multiple hypergeometric series – Appell series and beyond we have c, c − a − b0 a, b F a; b, b0; c; x, 1 = Γ F ; x , 1 c − a, c − b0 2 1 c − b0
for <(c − a − b0) > 0.
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F1[y = 1]: Since 0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y (1)m (c)m c + m m≥0
and a, b c, c − a − b F ; 1 = Γ , <(c − a − b) > 0, 2 1 c c − a, c − b
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
F1[y = 1]: Since 0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y (1)m (c)m c + m m≥0
and a, b c, c − a − b F ; 1 = Γ , <(c − a − b) > 0, 2 1 c c − a, c − b
we have c, c − a − b0 a, b F a; b, b0; c; x, 1 = Γ F ; x , 1 c − a, c − b0 2 1 c − b0
for <(c − a − b0) > 0.
Multiple hypergeometric series – Appell series and beyond we have 0 0 −b0 X (a)m (b)m m c − a, b y F1 a; b, b ; c; x, y = (1 − y) x 2F1 ; (1)m (c)m c + m y − 1 m≥0 0 y = (1 − y)−b F a, c − a; b, b0; c; x, . 3 y − 1
Hence, any F1 function can be expressed in terms of an F3 function. The converse is only true when c = a + a0.
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Since 0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y (1)m (c)m c + m m≥0
and a, b c − a, b y F ; y = (1 − y)−b F ; , 2 1 c 2 1 c y − 1
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Since 0 0 X (a)m (b)m m a + m, b F1 a; b, b ; c; x, y = x 2F1 ; y (1)m (c)m c + m m≥0
and a, b c − a, b y F ; y = (1 − y)−b F ; , 2 1 c 2 1 c y − 1 we have 0 0 −b0 X (a)m (b)m m c − a, b y F1 a; b, b ; c; x, y = (1 − y) x 2F1 ; (1)m (c)m c + m y − 1 m≥0 0 y = (1 − y)−b F a, c − a; b, b0; c; x, . 3 y − 1
Hence, any F1 function can be expressed in terms of an F3 function. The converse is only true when c = a + a0.
Multiple hypergeometric series – Appell series and beyond 0 Similarly, any F2 function reduces to an F1 function when c = a: 0 x F a; b, b0; c, a; x, y = (1 − y)−b F b; a − b0, b0; c; x, . 2 1 1 − y
(Conversely, any F1 function can be expressed in terms of an F2 function where c0 = a.) If further c = a, then
0 0 b, b xy F a; b, b0; a, a; x, y = (1 − x)−b(1 − y)−b F ; . 2 2 1 a (1 − x)(1 − y)
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Since the F1 function reduces to an ordinary 2F1 function when c = b + b0, we have
y F a, c − a; b, c − b; c; x, 3 y − 1 a, c − b y − x = (1 − x)−a(1 − y)c−b F ; . 2 1 c 1 − x
Multiple hypergeometric series – Appell series and beyond If further c = a, then
0 0 b, b xy F a; b, b0; a, a; x, y = (1 − x)−b(1 − y)−b F ; . 2 2 1 a (1 − x)(1 − y)
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Since the F1 function reduces to an ordinary 2F1 function when c = b + b0, we have
y F a, c − a; b, c − b; c; x, 3 y − 1 a, c − b y − x = (1 − x)−a(1 − y)c−b F ; . 2 1 c 1 − x
0 Similarly, any F2 function reduces to an F1 function when c = a: 0 x F a; b, b0; c, a; x, y = (1 − y)−b F b; a − b0, b0; c; x, . 2 1 1 − y
(Conversely, any F1 function can be expressed in terms of an F2 function where c0 = a.)
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Since the F1 function reduces to an ordinary 2F1 function when c = b + b0, we have
y F a, c − a; b, c − b; c; x, 3 y − 1 a, c − b y − x = (1 − x)−a(1 − y)c−b F ; . 2 1 c 1 − x
0 Similarly, any F2 function reduces to an F1 function when c = a: 0 x F a; b, b0; c, a; x, y = (1 − y)−b F b; a − b0, b0; c; x, . 2 1 1 − y
(Conversely, any F1 function can be expressed in terms of an F2 function where c0 = a.) If further c = a, then
0 0 b, b xy F a; b, b0; a, a; x, y = (1 − x)−b(1 − y)−b F ; . 2 2 1 a (1 − x)(1 − y)
Multiple hypergeometric series – Appell series and beyond The c0 = 1 + a + b − c special case gives the product formula F4 a; b; c, 1 + a + b − c; x(1 − y), y(1 − x) a, b a, b = F ; x F ; y . 2 1 c 2 1 c0
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
J.L Burchnall& T.W. Chaundy, 1940, 1941:
0 F4 a; b; c, c ; x(1 − y), y(1 − x) 0 X (a)m (b)m (1 + a + b − c − c )m m m = 0 x y m!(c)m (c )m m≥0 a + m, b + m a + m, b + m × F ; x F ; y . 2 1 c + m 2 1 c0 + m
This expansion has applications to classical orthogonal polynomials. It can also be used to deduce the double integral representation for F4.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
J.L Burchnall& T.W. Chaundy, 1940, 1941:
0 F4 a; b; c, c ; x(1 − y), y(1 − x) 0 X (a)m (b)m (1 + a + b − c − c )m m m = 0 x y m!(c)m (c )m m≥0 a + m, b + m a + m, b + m × F ; x F ; y . 2 1 c + m 2 1 c0 + m
This expansion has applications to classical orthogonal polynomials. It can also be used to deduce the double integral representation for F4.
The c0 = 1 + a + b − c special case gives the product formula F4 a; b; c, 1 + a + b − c; x(1 − y), y(1 − x) a, b a, b = F ; x F ; y . 2 1 c 2 1 c0
Multiple hypergeometric series – Appell series and beyond while putting in addition c = a gives the attractive summation formula
1−b 1−a −1 F4 a; b; a, b; x(1 − y), y(1 − x) = (1 − x) (1 − y) (1 − x − y) . Written out in explicit terms, this is
X X (a)m+n (b)m+n x m(1 − y)my n(1 − x)n m! n!(a)m (b)n m≥0 n≥0 = (1 − x)1−b(1 − y)1−a(1 − x − y)−1.
For y = 0 this reduces to Newton’s binomial expansion formula b F ; x = (1 − x)−b. 1 0 −
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums On the other hand, the c0 = b special case gives the reduction formula F4 a; b; c, b; x(1 − y), y(1 − x) xy x = (1 − x)−a(1 − y)−a F a; 1 + a − c, c − b; c; , , 1 (1 − x)(1 − y) x − 1
Multiple hypergeometric series – Appell series and beyond For y = 0 this reduces to Newton’s binomial expansion formula b F ; x = (1 − x)−b. 1 0 −
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums On the other hand, the c0 = b special case gives the reduction formula F4 a; b; c, b; x(1 − y), y(1 − x) xy x = (1 − x)−a(1 − y)−a F a; 1 + a − c, c − b; c; , , 1 (1 − x)(1 − y) x − 1
while putting in addition c = a gives the attractive summation formula
1−b 1−a −1 F4 a; b; a, b; x(1 − y), y(1 − x) = (1 − x) (1 − y) (1 − x − y) . Written out in explicit terms, this is
X X (a)m+n (b)m+n x m(1 − y)my n(1 − x)n m! n!(a)m (b)n m≥0 n≥0 = (1 − x)1−b(1 − y)1−a(1 − x − y)−1.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums On the other hand, the c0 = b special case gives the reduction formula F4 a; b; c, b; x(1 − y), y(1 − x) xy x = (1 − x)−a(1 − y)−a F a; 1 + a − c, c − b; c; , , 1 (1 − x)(1 − y) x − 1
while putting in addition c = a gives the attractive summation formula
1−b 1−a −1 F4 a; b; a, b; x(1 − y), y(1 − x) = (1 − x) (1 − y) (1 − x − y) . Written out in explicit terms, this is
X X (a)m+n (b)m+n x m(1 − y)my n(1 − x)n m! n!(a)m (b)n m≥0 n≥0 = (1 − x)1−b(1 − y)1−a(1 − x − y)−1.
For y = 0 this reduces to Newton’s binomial expansion formula b F ; x = (1 − x)−b. 1 0 −
Multiple hypergeometric series – Appell series and beyond In 1931, Jacob Horn studied 34 different convergent bivariate hypergeometric series (among which are F1, F2, F3, F4).
In 1937, M.-J. Kamp´ede F´eriet introduced the following bivariate extension of the generalized hypergeometric series:
0 0 p:q a1,..., ap : b1, b1; ... ; bq, bq; Fr:s 0 0 x, y c1,..., cr : d1, d1; ... ; ds , ds ; 0 0 m n X X (a1)m+n ... (ap)m+n (b1)m(b1)n ... (bq)m(bq)n x y = 0 0 . (c1)m+n ... (cr )m+n (d1)m(d )n ... (ds )m(d )n m!n! m≥0 n≥0 1 s
Numerous identities exist for special instances of such series.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Extensions
Multiple hypergeometric series – Appell series and beyond In 1937, M.-J. Kamp´ede F´eriet introduced the following bivariate extension of the generalized hypergeometric series:
0 0 p:q a1,..., ap : b1, b1; ... ; bq, bq; Fr:s 0 0 x, y c1,..., cr : d1, d1; ... ; ds , ds ; 0 0 m n X X (a1)m+n ... (ap)m+n (b1)m(b1)n ... (bq)m(bq)n x y = 0 0 . (c1)m+n ... (cr )m+n (d1)m(d )n ... (ds )m(d )n m!n! m≥0 n≥0 1 s
Numerous identities exist for special instances of such series.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Extensions
In 1931, Jacob Horn studied 34 different convergent bivariate hypergeometric series (among which are F1, F2, F3, F4).
Multiple hypergeometric series – Appell series and beyond Numerous identities exist for special instances of such series.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Extensions
In 1931, Jacob Horn studied 34 different convergent bivariate hypergeometric series (among which are F1, F2, F3, F4).
In 1937, M.-J. Kamp´ede F´eriet introduced the following bivariate extension of the generalized hypergeometric series:
0 0 p:q a1,..., ap : b1, b1; ... ; bq, bq; Fr:s 0 0 x, y c1,..., cr : d1, d1; ... ; ds , ds ; 0 0 m n X X (a1)m+n ... (ap)m+n (b1)m(b1)n ... (bq)m(bq)n x y = 0 0 . (c1)m+n ... (cr )m+n (d1)m(d )n ... (ds )m(d )n m!n! m≥0 n≥0 1 s
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Extensions
In 1931, Jacob Horn studied 34 different convergent bivariate hypergeometric series (among which are F1, F2, F3, F4).
In 1937, M.-J. Kamp´ede F´eriet introduced the following bivariate extension of the generalized hypergeometric series:
0 0 p:q a1,..., ap : b1, b1; ... ; bq, bq; Fr:s 0 0 x, y c1,..., cr : d1, d1; ... ; ds , ds ; 0 0 m n X X (a1)m+n ... (ap)m+n (b1)m(b1)n ... (bq)m(bq)n x y = 0 0 . (c1)m+n ... (cr )m+n (d1)m(d )n ... (ds )m(d )n m!n! m≥0 n≥0 1 s
Numerous identities exist for special instances of such series.
Multiple hypergeometric series – Appell series and beyond S.N. Pitre and J. Van der Jeugt, 1996:
− : a, d − a; b, d − b; c, d − c; e, e + d − a − b − c, e − d F 0:3 1, 1 = Γ , 1:1 d : e, d + e − a − b − c; e − a, e − b, e − c
where <(e − d) > 0 and <(d + e − a − b − c) > 0.
− : a, d − a; b, d − b; c, e − c − 1; F 0:3 1, 1 1:1 d : e, d + e − a − b − c; 1 − a, 1 − b, e, e − d, d + e − a − b − c = Γ , 1 − d, e − a, e − b, e − c, 1 + d − a − b
where <(d + e − a − b − c) > 0, and d − a or d − b is a negative integer.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
P.W. Karlsson, 1994:
− : a, d − a; b, d − b; c, −c; e, e + d − a − b − c F 0:3 1, 1 = Γ , 1:1 d : e, d + e − a − b − c; e − c, e + d − a − b
where <(e) > 0 and <(d + e − a − b − c) > 0.
Multiple hypergeometric series – Appell series and beyond − : a, d − a; b, d − b; c, e − c − 1; F 0:3 1, 1 1:1 d : e, d + e − a − b − c; 1 − a, 1 − b, e, e − d, d + e − a − b − c = Γ , 1 − d, e − a, e − b, e − c, 1 + d − a − b
where <(d + e − a − b − c) > 0, and d − a or d − b is a negative integer.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
P.W. Karlsson, 1994:
− : a, d − a; b, d − b; c, −c; e, e + d − a − b − c F 0:3 1, 1 = Γ , 1:1 d : e, d + e − a − b − c; e − c, e + d − a − b
where <(e) > 0 and <(d + e − a − b − c) > 0.
S.N. Pitre and J. Van der Jeugt, 1996:
− : a, d − a; b, d − b; c, d − c; e, e + d − a − b − c, e − d F 0:3 1, 1 = Γ , 1:1 d : e, d + e − a − b − c; e − a, e − b, e − c
where <(e − d) > 0 and <(d + e − a − b − c) > 0.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
P.W. Karlsson, 1994:
− : a, d − a; b, d − b; c, −c; e, e + d − a − b − c F 0:3 1, 1 = Γ , 1:1 d : e, d + e − a − b − c; e − c, e + d − a − b
where <(e) > 0 and <(d + e − a − b − c) > 0.
S.N. Pitre and J. Van der Jeugt, 1996:
− : a, d − a; b, d − b; c, d − c; e, e + d − a − b − c, e − d F 0:3 1, 1 = Γ , 1:1 d : e, d + e − a − b − c; e − a, e − b, e − c
where <(e − d) > 0 and <(d + e − a − b − c) > 0.
− : a, d − a; b, d − b; c, e − c − 1; F 0:3 1, 1 1:1 d : e, d + e − a − b − c; 1 − a, 1 − b, e, e − d, d + e − a − b − c = Γ , 1 − d, e − a, e − b, e − c, 1 + d − a − b
where <(d + e − a − b − c) > 0, and d − a or d − b is a negative integer.
Multiple hypergeometric series – Appell series and beyond (n) FA a; b1,..., bn; c1,..., cn; x1,..., xn (a) (b ) ... (b ) X X m1+···+mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c1)m1 ... (cn)mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1| + ··· + |xn| < 1.
(n) FB a1,..., an; b1,..., bn; c; x1,..., xn (a ) ... (a ) (b ) ... (b ) X X 1 m1 n mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c)m1+···+mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1|,..., |xn| < 1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1893, Giuseppe Lauricella investigated properties of the following four (n) (n) (n) (n) series FA , FB , FC , FD , of n variables:
Multiple hypergeometric series – Appell series and beyond (n) FB a1,..., an; b1,..., bn; c; x1,..., xn (a ) ... (a ) (b ) ... (b ) X X 1 m1 n mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c)m1+···+mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1|,..., |xn| < 1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1893, Giuseppe Lauricella investigated properties of the following four (n) (n) (n) (n) series FA , FB , FC , FD , of n variables:
(n) FA a; b1,..., bn; c1,..., cn; x1,..., xn (a) (b ) ... (b ) X X m1+···+mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c1)m1 ... (cn)mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1| + ··· + |xn| < 1.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1893, Giuseppe Lauricella investigated properties of the following four (n) (n) (n) (n) series FA , FB , FC , FD , of n variables:
(n) FA a; b1,..., bn; c1,..., cn; x1,..., xn (a) (b ) ... (b ) X X m1+···+mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c1)m1 ... (cn)mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1| + ··· + |xn| < 1.
(n) FB a1,..., an; b1,..., bn; c; x1,..., xn (a ) ... (a ) (b ) ... (b ) X X 1 m1 n mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c)m1+···+mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1|,..., |xn| < 1.
Multiple hypergeometric series – Appell series and beyond (n) FD a; b1,..., bn; c; x1,..., xn (a) (b ) ... (b ) X X m1+···+mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c)m1+···+mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1|,..., |xn| < 1. We have
(2) (2) (2) (2) FA = F2, FB = F3, FC = F4, FD = F1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
(n) FC a; b; c1,..., cn; x1,..., xn (a) (b) X X m1+···+mn m1+···+mn m1 mn = ··· x1 ... xn , (c1)m1 ... (cn)mn (1)m1 ... (1)mn m1≥0 mn≥0
1 1 where |x1| 2 + ··· + |xn| 2 < 1.
Multiple hypergeometric series – Appell series and beyond We have
(2) (2) (2) (2) FA = F2, FB = F3, FC = F4, FD = F1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
(n) FC a; b; c1,..., cn; x1,..., xn (a) (b) X X m1+···+mn m1+···+mn m1 mn = ··· x1 ... xn , (c1)m1 ... (cn)mn (1)m1 ... (1)mn m1≥0 mn≥0
1 1 where |x1| 2 + ··· + |xn| 2 < 1.
(n) FD a; b1,..., bn; c; x1,..., xn (a) (b ) ... (b ) X X m1+···+mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c)m1+···+mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1|,..., |xn| < 1.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
(n) FC a; b; c1,..., cn; x1,..., xn (a) (b) X X m1+···+mn m1+···+mn m1 mn = ··· x1 ... xn , (c1)m1 ... (cn)mn (1)m1 ... (1)mn m1≥0 mn≥0
1 1 where |x1| 2 + ··· + |xn| 2 < 1.
(n) FD a; b1,..., bn; c; x1,..., xn (a) (b ) ... (b ) X X m1+···+mn 1 m1 n mn m1 mn = ··· x1 ... xn , (c)m1+···+mn (1)m1 ... (1)mn m1≥0 mn≥0
where |x1|,..., |xn| < 1. We have
(2) (2) (2) (2) FA = F2, FB = F3, FC = F4, FD = F1.
Multiple hypergeometric series – Appell series and beyond (n) FD a; b1,..., bn; c; x1,..., xn Z 1 c a−1 c−a−1 −b1 −bn = Γ u (1 − u) (1 − ux1) ... (1 − uxn) du, a, c − a 0
where
This can be easily verified by Taylor expansion of the integrand, followed by termwise integration.
Further important extension (not considered here): Multivariate hypergeometric functions in the sense of I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinky, late 1980’s.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums (n) Integral representation of FD
Multiple hypergeometric series – Appell series and beyond This can be easily verified by Taylor expansion of the integrand, followed by termwise integration.
Further important extension (not considered here): Multivariate hypergeometric functions in the sense of I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinky, late 1980’s.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums (n) Integral representation of FD
(n) FD a; b1,..., bn; c; x1,..., xn Z 1 c a−1 c−a−1 −b1 −bn = Γ u (1 − u) (1 − ux1) ... (1 − uxn) du, a, c − a 0
where
Multiple hypergeometric series – Appell series and beyond Further important extension (not considered here): Multivariate hypergeometric functions in the sense of I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinky, late 1980’s.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums (n) Integral representation of FD
(n) FD a; b1,..., bn; c; x1,..., xn Z 1 c a−1 c−a−1 −b1 −bn = Γ u (1 − u) (1 − ux1) ... (1 − uxn) du, a, c − a 0
where
This can be easily verified by Taylor expansion of the integrand, followed by termwise integration.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums (n) Integral representation of FD
(n) FD a; b1,..., bn; c; x1,..., xn Z 1 c a−1 c−a−1 −b1 −bn = Γ u (1 − u) (1 − ux1) ... (1 − uxn) du, a, c − a 0
where
This can be easily verified by Taylor expansion of the integrand, followed by termwise integration.
Further important extension (not considered here): Multivariate hypergeometric functions in the sense of I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinky, late 1980’s.
Multiple hypergeometric series – Appell series and beyond In 2004, George Gasper and I derived (by specializing some curious q-series identities obtained by inverse relations) the following curious beta integral evaluations:
Z 1 β β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c − (a + 1) ) 2 2β 2 Γ(2β) 0 (c − (a + t) ) × tβ (1 − t)β−1 dt
and
Z 1 β−1 β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c−(a+1) ) 2 2β Γ(2β) 0 (c − (a + t) ) × (c − (a − t)(a + t)) tβ−1 (1 − t)β−1 dt,
where <(β) > 0. In our paper, we claimed that these integrals would be difficult to prove with standard methods.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums An application: Mizan Rahman’s evaluation of an integral
Multiple hypergeometric series – Appell series and beyond Z 1 β β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c − (a + 1) ) 2 2β 2 Γ(2β) 0 (c − (a + t) ) × tβ (1 − t)β−1 dt
and
Z 1 β−1 β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c−(a+1) ) 2 2β Γ(2β) 0 (c − (a + t) ) × (c − (a − t)(a + t)) tβ−1 (1 − t)β−1 dt,
where <(β) > 0. In our paper, we claimed that these integrals would be difficult to prove with standard methods.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums An application: Mizan Rahman’s evaluation of an integral
In 2004, George Gasper and I derived (by specializing some curious q-series identities obtained by inverse relations) the following curious beta integral evaluations:
Multiple hypergeometric series – Appell series and beyond In our paper, we claimed that these integrals would be difficult to prove with standard methods.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums An application: Mizan Rahman’s evaluation of an integral
In 2004, George Gasper and I derived (by specializing some curious q-series identities obtained by inverse relations) the following curious beta integral evaluations:
Z 1 β β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c − (a + 1) ) 2 2β 2 Γ(2β) 0 (c − (a + t) ) × tβ (1 − t)β−1 dt
and
Z 1 β−1 β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c−(a+1) ) 2 2β Γ(2β) 0 (c − (a + t) ) × (c − (a − t)(a + t)) tβ−1 (1 − t)β−1 dt,
where <(β) > 0.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums An application: Mizan Rahman’s evaluation of an integral
In 2004, George Gasper and I derived (by specializing some curious q-series identities obtained by inverse relations) the following curious beta integral evaluations:
Z 1 β β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c − (a + 1) ) 2 2β 2 Γ(2β) 0 (c − (a + t) ) × tβ (1 − t)β−1 dt
and
Z 1 β−1 β−1 Γ(β)Γ(β) 2 (c − a(a + t)) (c − (a + 1)(a + t)) = (c−(a+1) ) 2 2β Γ(2β) 0 (c − (a + t) ) × (c − (a − t)(a + t)) tβ−1 (1 − t)β−1 dt,
where <(β) > 0. In our paper, we claimed that these integrals would be difficult to prove with standard methods.
Multiple hypergeometric series – Appell series and beyond At some point, Mizan Rahman’s proof uses a reduction formula for an F1 Appell series.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Mizan Rahman, after seeing our paper, did not agree with our claim. He immediately sent us a fax with an “elementary proof” of the first identity using standard machinery.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Mizan Rahman, after seeing our paper, did not agree with our claim. He immediately sent us a fax with an “elementary proof” of the first identity using standard machinery.
At some point, Mizan Rahman’s proof uses a reduction formula for an F1 Appell series.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Mizan Rahman, after seeing our paper, did not agree with our claim. He immediately sent us a fax with an “elementary proof” of the first identity using standard machinery.
At some point, Mizan Rahman’s proof uses a reduction formula for an F1 Appell series.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Multiple hypergeometric series – Appell series and beyond We denote the q-shifted factorial by
k−1 (a; q)0 = 1, (a; q)k = (1 − a)(1 − aq) ... (1 − aq ), (a1,..., am; q)k = (a1; q)k ... (am; q)k , k ∈ N ∪ ∞. The continuous q-ultraspherical polynomials are given by
n X (β; q)k (β; q)n−k i(n−2k)θ Cn(x; β|q) = e , x = cos θ. (q; q)k (q; q)n−k k=0
They satisfy (for |q|, |β| < 1) the orthogonality relation
Z 1 2iθ −2iθ 1 (e , e ; q)∞ dx Cm(x; β|q) Cn(x; β|q) √ 2iθ −2iθ 2 2π −1 (βe , βe ; q)∞ 1 − x 2 (β, βq; q)∞ (β ; q)n (1 − β) = 2 n δm,n. (q, β ; q)∞ (q; q)n (1 − βq )
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums More sums
Multiple hypergeometric series – Appell series and beyond The continuous q-ultraspherical polynomials are given by
n X (β; q)k (β; q)n−k i(n−2k)θ Cn(x; β|q) = e , x = cos θ. (q; q)k (q; q)n−k k=0
They satisfy (for |q|, |β| < 1) the orthogonality relation
Z 1 2iθ −2iθ 1 (e , e ; q)∞ dx Cm(x; β|q) Cn(x; β|q) √ 2iθ −2iθ 2 2π −1 (βe , βe ; q)∞ 1 − x 2 (β, βq; q)∞ (β ; q)n (1 − β) = 2 n δm,n. (q, β ; q)∞ (q; q)n (1 − βq )
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums More sums
We denote the q-shifted factorial by
k−1 (a; q)0 = 1, (a; q)k = (1 − a)(1 − aq) ... (1 − aq ), (a1,..., am; q)k = (a1; q)k ... (am; q)k , k ∈ N ∪ ∞.
Multiple hypergeometric series – Appell series and beyond They satisfy (for |q|, |β| < 1) the orthogonality relation
Z 1 2iθ −2iθ 1 (e , e ; q)∞ dx Cm(x; β|q) Cn(x; β|q) √ 2iθ −2iθ 2 2π −1 (βe , βe ; q)∞ 1 − x 2 (β, βq; q)∞ (β ; q)n (1 − β) = 2 n δm,n. (q, β ; q)∞ (q; q)n (1 − βq )
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums More sums
We denote the q-shifted factorial by
k−1 (a; q)0 = 1, (a; q)k = (1 − a)(1 − aq) ... (1 − aq ), (a1,..., am; q)k = (a1; q)k ... (am; q)k , k ∈ N ∪ ∞. The continuous q-ultraspherical polynomials are given by
n X (β; q)k (β; q)n−k i(n−2k)θ Cn(x; β|q) = e , x = cos θ. (q; q)k (q; q)n−k k=0
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums More sums
We denote the q-shifted factorial by
k−1 (a; q)0 = 1, (a; q)k = (1 − a)(1 − aq) ... (1 − aq ), (a1,..., am; q)k = (a1; q)k ... (am; q)k , k ∈ N ∪ ∞. The continuous q-ultraspherical polynomials are given by
n X (β; q)k (β; q)n−k i(n−2k)θ Cn(x; β|q) = e , x = cos θ. (q; q)k (q; q)n−k k=0
They satisfy (for |q|, |β| < 1) the orthogonality relation
Z 1 2iθ −2iθ 1 (e , e ; q)∞ dx Cm(x; β|q) Cn(x; β|q) √ 2iθ −2iθ 2 2π −1 (βe , βe ; q)∞ 1 − x 2 (β, βq; q)∞ (β ; q)n (1 − β) = 2 n δm,n. (q, β ; q)∞ (q; q)n (1 − βq )
Multiple hypergeometric series – Appell series and beyond For simplicity, write this as
X k Cm(x; β|q) Cn(x; β|q) = fm,n Ck (x; β|q), k
k k with explicity determined structure coefficients fm,n = fm,n(β, q). k k By definition, fm,n = fn,m.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1895, L.J. Rogers derived the following linearization formula for the continuous q-ultraspherical polynomials:
Cm(x; β|q) Cn(x; β|q)
min(m,n) 2 X (q; q)m+n−2k (β; q)m−k (β; q)n−k (β; q)k (β ; q)m+n−k = 2 (β ; q)m+n−2k (q; q)m−k (q; q)n−k (q; q)k (βq; q)m+n−k k=0 (1 − βqm+n−2k ) × C (x; β|q). (1 − β) m+n−2k
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
In 1895, L.J. Rogers derived the following linearization formula for the continuous q-ultraspherical polynomials:
Cm(x; β|q) Cn(x; β|q)
min(m,n) 2 X (q; q)m+n−2k (β; q)m−k (β; q)n−k (β; q)k (β ; q)m+n−k = 2 (β ; q)m+n−2k (q; q)m−k (q; q)n−k (q; q)k (βq; q)m+n−k k=0 (1 − βqm+n−2k ) × C (x; β|q). (1 − β) m+n−2k
For simplicity, write this as
X k Cm(x; β|q) Cn(x; β|q) = fm,n Ck (x; β|q), k
k k with explicity determined structure coefficients fm,n = fm,n(β, q). k k By definition, fm,n = fn,m.
Multiple hypergeometric series – Appell series and beyond Now taking coefficients of Cj (x; β|q) gives the transformation formula
X j k X j k fl,k fm,n = fn,k fm,l . k k
More generally, the r-fold sum
X k f j f k1 f k2 ... f r−1 f kr m0,k1 m1,k2 m2,k3 mr−1,kr mr ,mr+1 k1,...,kr
is symmetric in {m0, m1,..., mr+1}, resulting in transformation formulae for multivariate basic hypergeometric series.
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Linearization of the triple product Cl (x; β|q) Cm(x; β|q) Cn(x; β|q) in two different ways gives
X X j k X X j k fl,k fm,n Cj (x; β|q) = fn,k fm,l Cj (x; β|q), j k j k
as we must have symmetry between n and l.
Multiple hypergeometric series – Appell series and beyond More generally, the r-fold sum
X k f j f k1 f k2 ... f r−1 f kr m0,k1 m1,k2 m2,k3 mr−1,kr mr ,mr+1 k1,...,kr
is symmetric in {m0, m1,..., mr+1}, resulting in transformation formulae for multivariate basic hypergeometric series.
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Linearization of the triple product Cl (x; β|q) Cm(x; β|q) Cn(x; β|q) in two different ways gives
X X j k X X j k fl,k fm,n Cj (x; β|q) = fn,k fm,l Cj (x; β|q), j k j k
as we must have symmetry between n and l.
Now taking coefficients of Cj (x; β|q) gives the transformation formula
X j k X j k fl,k fm,n = fn,k fm,l . k k
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Linearization of the triple product Cl (x; β|q) Cm(x; β|q) Cn(x; β|q) in two different ways gives
X X j k X X j k fl,k fm,n Cj (x; β|q) = fn,k fm,l Cj (x; β|q), j k j k
as we must have symmetry between n and l.
Now taking coefficients of Cj (x; β|q) gives the transformation formula
X j k X j k fl,k fm,n = fn,k fm,l . k k
More generally, the r-fold sum
X k f j f k1 f k2 ... f r−1 f kr m0,k1 m1,k2 m2,k3 mr−1,kr mr ,mr+1 k1,...,kr
is symmetric in {m0, m1,..., mr+1}, resulting in transformation formulae for multivariate basic hypergeometric series.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
The r = 1 case, after analytic continuation can be written as the following transformation formula for a very-well-poised 14φ13 series (R. Langer, MJS& S.O. Warnaar, 2009):
n 2k X(1 − aq ) (aq/b; q)2k (1 − a) (ab; q)2k k=0 n −n 2k (a, b, c, d, ab/c, ab/d, abq , q ; q)k q × 1−n n+1 (q, aq/b, aq/c, aq/d, cq/b, dq/b, q /b, aq ; q)k b n 2k (aq, ˆaq/c, ˆaq/d, aq/cd; q)n X (1 − ˆaq ) (ˆaq/b; q)2k = (ˆaq, aq/c, aq/d, ˆaq/cd; q)n (1 − ˆa) (ˆab; q)2k k=0 n −n 2k (ˆa, b, c, d, ˆab/c, ˆab/d, ˆabq , q ; q)k q × 1−n n+1 , (q, ˆaq/b, ˆaq/c, ˆaq/d, cq/b, dq/b, q /b, ˆaq ; q)k b
where ˆa = q−ncd/ab.
Multiple hypergeometric series – Appell series and beyond These identities can be extended to the elliptic setting. For the latter, we have 2l+2k m −m X θ(abq ; p) (aq , q ; q, p)l+k (b, ab/c, ab/d, aq/cd; q, p)l l 1−m m+1 q θ(ab; p) (bq , abq ; q, p)l+k (q, aq/c, aq/d, ab/cd; q, p)l l,k≥0 k−l 1−l 2 θ(cdq /ab; p) (cdq /ab ; q, p)k (b, c, d, cd/a; q, p)k k × −l −l q θ(cdq /ab; p) (cdq /a; q, p)k (q, cq/b, dq/b, cdq/ab; q, p)k
m (aq/b; q, p)2m (abq, b, c, d, ab/c, ab/d; q, p)m q = 2 . (ab; q, p)2m (1/b, aq/b, aq/c, aq/d, cq/b, dq/b; q, p)m b
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums By inverse relations, one obtains the following double sum identity: 2l+2k m −m X (1 − abq ) (aq , q ; q)l+k (b, ab/c, ab/d, aq/cd; q)l l 1−m m+1 q (1 − ab) (bq , abq ; q)l+k (q, aq/c, aq/d, ab/cd; q)l l,k≥0 k−l 1−l 2 (1 − cdq /ab) (cdq /ab ; q)k (b, c, d, cd/a; q)k k × −l −l q (1 − cdq /ab) (cdq /a; q)k (q, cq/b, dq/b, cdq/ab; q)k
m (aq/b; q)2m (abq, b, c, d, ab/c, ab/d; q)m q = 2 . (ab; q)2m (1/b, aq/b, aq/c, aq/d, cq/b, dq/b; q)m b
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums By inverse relations, one obtains the following double sum identity: 2l+2k m −m X (1 − abq ) (aq , q ; q)l+k (b, ab/c, ab/d, aq/cd; q)l l 1−m m+1 q (1 − ab) (bq , abq ; q)l+k (q, aq/c, aq/d, ab/cd; q)l l,k≥0 k−l 1−l 2 (1 − cdq /ab) (cdq /ab ; q)k (b, c, d, cd/a; q)k k × −l −l q (1 − cdq /ab) (cdq /a; q)k (q, cq/b, dq/b, cdq/ab; q)k
m (aq/b; q)2m (abq, b, c, d, ab/c, ab/d; q)m q = 2 . (ab; q)2m (1/b, aq/b, aq/c, aq/d, cq/b, dq/b; q)m b
These identities can be extended to the elliptic setting. For the latter, we have 2l+2k m −m X θ(abq ; p) (aq , q ; q, p)l+k (b, ab/c, ab/d, aq/cd; q, p)l l 1−m m+1 q θ(ab; p) (bq , abq ; q, p)l+k (q, aq/c, aq/d, ab/cd; q, p)l l,k≥0 k−l 1−l 2 θ(cdq /ab; p) (cdq /ab ; q, p)k (b, c, d, cd/a; q, p)k k × −l −l q θ(cdq /ab; p) (cdq /a; q, p)k (q, cq/b, dq/b, cdq/ab; q, p)k
m (aq/b; q, p)2m (abq, b, c, d, ab/c, ab/d; q, p)m q = 2 . (ab; q, p)2m (1/b, aq/b, aq/c, aq/d, cq/b, dq/b; q, p)m b
Multiple hypergeometric series – Appell series and beyond Let |p| < 1. (Modified Jacobi) theta functions:
∞ Y j j+1 θ(x; p) := (x, p/x; p)∞ = (1 − p x)(1 − p /x). j=0
Theta shifted factorials:
k−1 (a; q, p)k := θ(a; p)θ(aq; p) ··· θ(aq ; p) for k = 0, 1, 2,....
There holds θ(x; 0) = (1 − x) and (a; q, 0)k = (a; q)k . Compact notations:
θ(x1,..., xm; p) := θ(x1; p) ··· θ(xm; p),
(a1,..., am; q; p)k := (a1; q, p)k ··· (am; q, p)k .
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Elliptic hypergeometric series
Multiple hypergeometric series – Appell series and beyond Theta shifted factorials:
k−1 (a; q, p)k := θ(a; p)θ(aq; p) ··· θ(aq ; p) for k = 0, 1, 2,....
There holds θ(x; 0) = (1 − x) and (a; q, 0)k = (a; q)k . Compact notations:
θ(x1,..., xm; p) := θ(x1; p) ··· θ(xm; p),
(a1,..., am; q; p)k := (a1; q, p)k ··· (am; q, p)k .
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Elliptic hypergeometric series
Let |p| < 1. (Modified Jacobi) theta functions:
∞ Y j j+1 θ(x; p) := (x, p/x; p)∞ = (1 − p x)(1 − p /x). j=0
Multiple hypergeometric series – Appell series and beyond Compact notations:
θ(x1,..., xm; p) := θ(x1; p) ··· θ(xm; p),
(a1,..., am; q; p)k := (a1; q, p)k ··· (am; q, p)k .
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Elliptic hypergeometric series
Let |p| < 1. (Modified Jacobi) theta functions:
∞ Y j j+1 θ(x; p) := (x, p/x; p)∞ = (1 − p x)(1 − p /x). j=0
Theta shifted factorials:
k−1 (a; q, p)k := θ(a; p)θ(aq; p) ··· θ(aq ; p) for k = 0, 1, 2,....
There holds θ(x; 0) = (1 − x) and (a; q, 0)k = (a; q)k .
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums Elliptic hypergeometric series
Let |p| < 1. (Modified Jacobi) theta functions:
∞ Y j j+1 θ(x; p) := (x, p/x; p)∞ = (1 − p x)(1 − p /x). j=0
Theta shifted factorials:
k−1 (a; q, p)k := θ(a; p)θ(aq; p) ··· θ(aq ; p) for k = 0, 1, 2,....
There holds θ(x; 0) = (1 − x) and (a; q, 0)k = (a; q)k . Compact notations:
θ(x1,..., xm; p) := θ(x1; p) ··· θ(xm; p),
(a1,..., am; q; p)k := (a1; q, p)k ··· (am; q, p)k .
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Inversion formula: 1 θ(1/x; p) = − θ(x; p). x
Quasi-periodicity: 1 θ(px; p) = − θ(x; p). x
Riemann relation: u θ(xy, x/y, uv, u/v; p) − θ(xv, x/v, uy, u/y; p) = θ(yv, y/v, xu, x/u; p). y
Multiple hypergeometric series – Appell series and beyond Without loss of generality,
x x x θ(a0q , a1q ,..., as q ) g(x) = 1+x x x z, θ(q , b1q ,..., bs q ) where a0a1 ··· as = qb1b2 ··· bs (elliptic balancing condition). If we write q = e2πiσ, p = e2πiτ , with complex σ, τ, then g(x) is periodic in x with periods σ−1 and τσ−1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Elliptic hypergeometric series:
X ck , k≥0
where c0 = 1 and g(k) = ck+1/ck is an elliptic (doubly periodic, meromorphic) function of k with k considered as a complex variable.
Multiple hypergeometric series – Appell series and beyond If we write q = e2πiσ, p = e2πiτ , with complex σ, τ, then g(x) is periodic in x with periods σ−1 and τσ−1.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Elliptic hypergeometric series:
X ck , k≥0
where c0 = 1 and g(k) = ck+1/ck is an elliptic (doubly periodic, meromorphic) function of k with k considered as a complex variable. Without loss of generality,
x x x θ(a0q , a1q ,..., as q ) g(x) = 1+x x x z, θ(q , b1q ,..., bs q ) where a0a1 ··· as = qb1b2 ··· bs (elliptic balancing condition).
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Elliptic hypergeometric series:
X ck , k≥0
where c0 = 1 and g(k) = ck+1/ck is an elliptic (doubly periodic, meromorphic) function of k with k considered as a complex variable. Without loss of generality,
x x x θ(a0q , a1q ,..., as q ) g(x) = 1+x x x z, θ(q , b1q ,..., bs q ) where a0a1 ··· as = qb1b2 ··· bs (elliptic balancing condition). If we write q = e2πiσ, p = e2πiτ , with complex σ, τ, then g(x) is periodic in x with periods σ−1 and τσ−1.
Multiple hypergeometric series – Appell series and beyond −n For convergence, one usually requires as = q (n being a nonnegative integer), so that the sum is finite.
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General solution: ∞ a0, a1,..., as X (a0, a1,..., as ; q, p)k k s+1Es ; q, p; z := z , b1, b2,..., bs (q, b1 ..., bs ; q, p)k k=0
where a0a1 ··· as = qb1b2 ··· bs .
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
General solution: ∞ a0, a1,..., as X (a0, a1,..., as ; q, p)k k s+1Es ; q, p; z := z , b1, b2,..., bs (q, b1 ..., bs ; q, p)k k=0
where a0a1 ··· as = qb1b2 ··· bs .
−n For convergence, one usually requires as = q (n being a nonnegative integer), so that the sum is finite.
Multiple hypergeometric series – Appell series and beyond Their systematic study commenced at about the turn of the millenium, after further pioneering work of Spiridonov and Zhedanov, and of Warnaar.
Frenkel and Turaev’s 10V9 summation:
n 2k −n X θ(aq ; p) (a, b, c, d, e, q ; q, p)k k n+1 q θ(a; p) (q, aq/b, aq/c, aq/d, aq/e, aq ; q, p)k k=0 (aq, aq/bc, aq/bd, aq/cd; q, p) = n , (aq/b, aq/c, aq/d, aq/bcd; q, p)n
where a2qn+1 = bcde.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Elliptic hypergeometric series first appeared as elliptic solutions of the Yang–Baxter equation in work by Date, Jimbo, Kuniba, Miwa and Okado in 1987, and ten years later by I. B. Frenkel and V. Turaev.
Multiple hypergeometric series – Appell series and beyond Frenkel and Turaev’s 10V9 summation:
n 2k −n X θ(aq ; p) (a, b, c, d, e, q ; q, p)k k n+1 q θ(a; p) (q, aq/b, aq/c, aq/d, aq/e, aq ; q, p)k k=0 (aq, aq/bc, aq/bd, aq/cd; q, p) = n , (aq/b, aq/c, aq/d, aq/bcd; q, p)n
where a2qn+1 = bcde.
Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Elliptic hypergeometric series first appeared as elliptic solutions of the Yang–Baxter equation in work by Date, Jimbo, Kuniba, Miwa and Okado in 1987, and ten years later by I. B. Frenkel and V. Turaev.
Their systematic study commenced at about the turn of the millenium, after further pioneering work of Spiridonov and Zhedanov, and of Warnaar.
Multiple hypergeometric series – Appell series and beyond Appell series Contiguous relations PDE’s Integral representations Transformations Reductions Extensions A curious integral More sums
Elliptic hypergeometric series first appeared as elliptic solutions of the Yang–Baxter equation in work by Date, Jimbo, Kuniba, Miwa and Okado in 1987, and ten years later by I. B. Frenkel and V. Turaev.
Their systematic study commenced at about the turn of the millenium, after further pioneering work of Spiridonov and Zhedanov, and of Warnaar.
Frenkel and Turaev’s 10V9 summation:
n 2k −n X θ(aq ; p) (a, b, c, d, e, q ; q, p)k k n+1 q θ(a; p) (q, aq/b, aq/c, aq/d, aq/e, aq ; q, p)k k=0 (aq, aq/bc, aq/bd, aq/cd; q, p) = n , (aq/b, aq/c, aq/d, aq/bcd; q, p)n
where a2qn+1 = bcde.
Multiple hypergeometric series – Appell series and beyond