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 ; jxj; jyj < 1: m! n!(c)m+n m≥0 n≥0 0 X X (a)m+n (b)m (b )n 0 0 m n jxj + jyj < 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 ; jxj; jyj < 1: m! n!(c)m+n m≥0 n≥0 0 X X (a)m+n (b)m+n m n 1 1 jxj 2 + jyj 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.