Generalizations of in Askey Schemes

Howard S. Cohl?

?Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Maryland Department of Mathematics and Statistics Colloqium Department of Mathematics and Statistics American University Washington, D.C.

April 15, 2014

Howard Cohl (NIST) Generalized expansions April 15, 2014 1 / 56 Acknowedgements (co-authors)

Generalized hypergeometric orthogonal polynomials Hans Volkmer, University of Wisconsin-Milwaukee Michael A. Baeder, student Harvey Mudd College

Basic hypergeometric orthogonal polynomials Roberto Costas-Santos, Universidad de Alcalá, Spain Philbert Hwang, student at Poolesville High School

Howard Cohl (NIST) Generalized expansions April 15, 2014 2 / 56 Orthogonal polynomials

Gabor Szeg®, Orthogonal Polynomials (1959) A polynomial is an expression of nite length constructed from variables and constants, i.e.,

2 4 n pn(z) = c0 + c1z + c2z + c3z + ··· + cnz

An orthogonal (orthonormal) set of polynomials {pn(x)} is dened by the relations

Z b pm(x)pn(x)w(x)dx = δm,n. a Askey scheme  a way of organizing orthogonal polynomials of hypergeometric type into a hierarchy

Howard Cohl (NIST) Generalized expansions April 15, 2014 3 / 56 Askey scheme

Howard Cohl (NIST) Generalized expansions April 15, 2014 4 / 56 Askey scheme Classical continuous orthogonal polynomials

Orthog. polynomial Symbol w(x) (a, b)

˛ ˛2 Wilson ˛ Γ(a+ix)Γ(b+ix)Γ(c+ix)Γ(d+ix) ˛ Wn ˛ Γ(2ix) ˛ (0, ∞) 2 Cts. dual Hahn S Γ(a+ix)Γ(b+ix)Γ(c+ix) (0, ∞) n Γ(2ix) 2 Continuous Hahn pn |Γ(a + ix)Γ(b + ix)| (−∞, ∞)

(λ) (2φ−π)x 2 Meixner-Pollaczek Pn e |Γ(λ + ix)| (−∞, ∞)

(α,β) α β Jacobi Pn (1 − x) (1 + x) (−1, 1)

−x α Laguerre Ln e x (0, ∞)

−x2 Hermite Hn e (−∞, ∞)

Howard Cohl (NIST) Generalized expansions April 15, 2014 5 / 56 Askey scheme Factorials and generalized hypergeometric series

Euler's gamma function and factorial for non-negative integers Z ∞ π Γ(z) := tz−1e−tdt, Re z > 0, Γ(z)Γ(1 − z) = 0 sin(πz)

Γ(n) = (n − 1)!, n ∈ N0 = 0, 1, 2,... Pochhammer symbol: the rising factorial in the complex plane

(a)n := (a)(a + 1) ... (a + n − 1), (a)0 := 1, a ∈ C Γ(a + n) (a) = , a 6∈ −N = 0, −1, −2,... n Γ(a) 0 Generalized hypergeometric series

  ∞ n a1, . . . , ar X (a1)n ... (ar)n z F ; z := r s b , . . . , b (b ) ... (b ) n! 1 s n=0 1 n s n

Howard Cohl (NIST) Generalized expansions April 15, 2014 6 / 56 Askey scheme Example: Denition of the (continuous) Wilson polynomials

No symmetry in the parameters

2 Wn(x ; a, b, c, d)  −n, n + a + b + c + d − 1, a + ix, a − ix  = (a + b) (a + c) (a + d) F ; 1 n n n 4 3 a + b, a + c, a + d

Symmetry in the parameters

2 (a − ix)n(b − ix)n(c − ix)n(d − ix)n Wn(x ; a, b, c, d) = (−2ix)n

1 ! −n, 2ix − n, ix − 2 n + 1, a + ix, b + ix, c + ix, d + ix × 7F6 1 ; 1 ix − 2 n, 1 − n − a + ix, 1 − n − b + ix, 1 − n − c + ix, 1 − n − d + ix

Howard Cohl (NIST) Generalized expansions April 15, 2014 7 / 56 q-Askey scheme

Howard Cohl (NIST) Generalized expansions April 15, 2014 8 / 56 q-Askey scheme q-calculus

q-Pochhammer symbol

n−1 (a; q)n := (1 − a)(1 − aq) ... (1 − aq ), (a; q)0 := 1 Notation

(a1, . . . , ar; q)n := (a1; q)n ... (ar; q)n Basic hypergeometric series

∞   1+s−r a1, . . . , ar X (a1, . . . , ar; q)n  n n  n φ ; q, z := (−1) q(2) z r s b , . . . , b (b , . . . , b , q; q) 1 s n=0 1 s n

  ∞ n a1, . . . , as+1 X (a1, . . . , as+1; q)n z φ ; q, z := s+1 s b , . . . , b (b , . . . , b ; q) (q; q) 1 s n=0 1 s n n

Howard Cohl (NIST) Generalized expansions April 15, 2014 9 / 56 Connection relations and generating functions Connection relations and coecients

n (α) X (β) Pn (x) = cn,k(α; β)Pk (x) k=0

What are the cn,k? This is a problem in orthogonal polynomials. In general, one can compute connection relations by using orthogonality

Z b (α) (α) Pk (x)Pk0 (x)w(x)dx = dk(α)δk,k0 . a Therefore

Z b 1 (α) (β) cn,k(α, β) = Pn (x)Pk (x)w(x)dx. dk(β) a

Howard Cohl (NIST) Generalized expansions April 15, 2014 10 / 56 Connection relations and generating functions Generating functions

∞ X n (α) f(x, ρ; α) = cn(α)ρ Pn (x) n=0 Examples: Hermite polynomials ∞ X 1 exp(2xρ − ρ2) = ρnH (x) n! n n=0 Gegenbauer polynomials ∞ 1 X = ρnCν(x) (1 + ρ2 − 2ρx)ν n n=0 Jacobi polynomials ∞ α+β −1 −α −β X n (α,β) 2 R (1 − ρ + R) (1 + ρ + R) = ρ Pn (x), n=0 where R = p1 + ρ2 − 2ρx. Howard Cohl (NIST) Generalized expansions April 15, 2014 11 / 56 Connection relations and generating functions Ex: Laguerre polynomial connection rel. via generating fn.

Denition (orthogonal, monic):

n k X (n + α)n−k (−x) L(α)(x) = n (n − k)! k! k=0 Generating function

∞  xρ  X (1 − ρ)−α−1 exp = ρnL(α)(x) ρ − 1 n n=0

∞ (1 − ρ)−α−1  xρ  X (1 − ρ)−β−1 exp = (1 − ρ)β−α ρnL(β)(x) (1 − ρ)−β−1 ρ − 1 n n=0 ∞ ∞ X (r)k X (α − β)j (1 − ρ)−r = xr−kyk =⇒ (1 − ρ)β−α = ρj k! j! k=0 j=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 12 / 56 Connection relations and generating functions Example: Laguerre polynomial (cont.)

∞ ∞ ∞ X (α − β)j X (β) X ρj ρkL (x) = ρnL(α)(x) j! k n j=0 k=0 n=0 j + k = n =⇒ j = n − k ∞ ( n ) X X (α − β)n−k (β) Lα(x) − L (x) = 0 n (n − k)! k n=0 k=0 Connection relation (1 free parameter) n (α) X (β) Ln (x) = cn,k(α; β) Lk (x), k=0 where (α − β) c (α; β) = n−k n,k (n − k)!

Howard Cohl (NIST) Generalized expansions April 15, 2014 13 / 56 Generalized expansions Generalizations of generating function families

Generalized hypergeometric orthogonal polynomials Laguerre polynomials Jacobi, Gegenbauer, Chebyshev and Legendre polynomials Wilson polynomials Continuous Hahn polynomials Continuous Meixner-Pollaczek polynomials Basic hypergeometric orthogonal polynomials q-ultraspherical Al-Salam-Carlitz II /Rogers polynomials big/little q-Jacobi polynomials (ongoing) q-Laguerre polynomials Stieltjes-Wigert polynomials (ongoing) big q-Laguerre Askey-Wilson (ongoing)

Howard Cohl (NIST) Generalized expansions April 15, 2014 14 / 56 Generalized expansions Example: Al-Salam-Carlitz II generating function

Denition (orthogonal, monic):  −n n  (α) n − n q , x q V (x; q) = (−α) q (2) φ ; q, , α < 0 n 2 0 − α Connection relation (1 free parameter) n (α) X (β) Vn (x; q) = cn,k(α; β) Vk (x; q). k=0 where n−k k(k−n) k−n+1 n−k+1 β q (q α/β; q)n−k(q ; q)k cn,k(α; β) := (q; q)k Generating function: ∞  x  X qn(n−1)ρn (αρ; q) φ ; q, ρ = V (α)(x; q) ∞ 1 1 αρ (q; q) n n=0 n

Howard Cohl (NIST) Generalized expansions April 15, 2014 15 / 56 Generalized expansions Example: Al-Salam-Carlitz II (cont.)

Therefore:

∞  x  X qn(n−1)ρn (αρ; q) φ ; q, ρ = ∞ 1 1 αρ (q; q) n=0 n

n n−k k(k−n) k−n+1 n−k+1 X β q (q α/β; q)n−k(q ; q)k (β) × Vk (x) (q; q)k k=0 Reverse the order of summation and let n 7→ n + k

  ∞ k (β) x X ρ Vk (x, q) (αρ; q)∞ 1φ1 ; q, ρ = αρ (q; q)k k=0

∞ n −nk 1−n n+1 (n+k)(n+k−1) n X β q (αq /β; q)n(q ; q)kq ρ × (q; q) n=0 n+k

Howard Cohl (NIST) Generalized expansions April 15, 2014 16 / 56 Generalized expansions Example: Al-Salam-Carlitz II (cont.)

Using the properties of q-Pochhammer symbols:

n (a; q)n+k = (a; q)n(aq , q)k,

−n n −n−(n) (aq ; q)n = (q/a; q)n(−a) q 2 , a 6= 0, one obtains after cancellation

∞ (β) ∞ n β   k k(k−1) (2) k n x X ρ q Vk (x, q) X q ( α ; q)n(−αρq ) (αρ; q)∞ 1φ1 ; q, ρ = αρ (q; q)k (q; q)n k=0 n=0 therefore with α, β < 0, one has

  ∞ n n(n−1) (β)   x X ρ q Vn (x, q) β/α (αρ; q) φ ; q, ρ = φ ; q, αρqn ∞ 1 1 αρ (q; q) 1 1 0 n=0 n

Howard Cohl (NIST) Generalized expansions April 15, 2014 17 / 56 Generalized expansions Classical expansions for orthogonal polynomials

Gegenbauer generating function (1874)

∞ 1 X = ρnCν(x) (1 + ρ2 − 2ρx)ν n n=0 Legendre generating function (1783)

∞ 1 X n = ρ Pn(x) p 2 1 + ρ − 2ρx n=0 Heine's reciprocal square root identity (1881)

√ ∞ 1 2 X  1 if n = 0, √ =  Q (z)T (x),  := z − x π n n−1/2 n n 2 if n = 1, 2, 3,... n=0 ∞ 1 1 X 1 + ρ2  = √ nQ Tn(x) p 2 π ρ n−1/2 2ρ 1 + ρ − 2ρx n=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 18 / 56 Generalized expansions Classical Gauss hypergeometric orthogonal polynomials

(α,β) Jacobi polynomials, α, β > −1,Pn : C → C, dened as −n, n + α + β + 1 ! (α,β) (α + 1)n 1 − z Pn (z) := 2F1 ; n! α + 1 2 Z 1 (α,β) (α,β) α β Pm (x)Pn (x)(1 − x) (1 + x) dx −1 2α+β+1Γ(α + n + 1)Γ(β + n + 1) = δ (2n + α + β + 1)Γ(α + β + n + 1)n! m,n

Gegenbauer polynomials, 1 µ dened µ ∈ (− 2 , ∞) \{0},Cn : C → C, as

−n, n + 2µ ! µ (2µ)n 1 − z (2µ)n (µ−1/2,µ−1/2) C (z) := 2F1 ; = P n n! 1 2 (µ + 1 ) n µ + 2 2 n

Howard Cohl (NIST) Generalized expansions April 15, 2014 19 / 56 Generalized expansions Classical Gauss hypergeometric orthogonal polynomials

Legendre polynomials: 1/2 Pn(z) = Cn (z) Chebyshev polynomials of the 1st kind:

Tn(cos θ) = cos(nθ)

1 n + µ µ Tn(z) = lim Cn (z) n µ→0 µ Chebyshev polynomials of the 2nd kind: 1 Un(z) = Cn(z)

Howard Cohl (NIST) Generalized expansions April 15, 2014 20 / 56 Generalized expansions Special functions: associated Legendre functions

µ Ferrers function of the rst kind: Pν :(−1, 1) → C (associated Legendre function of the rst kind on the cut)

1 1 + xµ/2  1 − x Pµ(x) := F −ν, ν + 1; 1 − µ; ν Γ(1 − µ) 1 − x 2 1 2 µ Legendre function of the rst kind: Pν : C \ (−∞, 1] → C 1 z + 1µ/2  1 − z  P µ(z) := F −ν, ν + 1; 1 − µ; ν Γ(1 − µ) z − 1 2 1 2 µ Legendre function of the second kind, Qν : C \ (−∞, 1] → C √ πeiπµΓ(ν + µ + 1)(z2 − 1)µ/2 Qµ(z) := ν ν+1 3 ν+µ+1 2 Γ(ν + 2 )z ν + µ + 2 ν + µ + 1 3 1  × F , ; ν + ; 2 1 2 2 2 z2

Howard Cohl (NIST) Generalized expansions April 15, 2014 21 / 56 Generalized expansions √ The alg. functions p1 + ρ2 − 2ρx, z − x from geometry

The distance between two points x, x0 ∈ Rd in a polyspherical coordinate system on Rd is given by q kx − x0k = r2 + r02 − 2rr0 cos γ, (1) x · x0 where r = kxk, r0 = kx0k, and cos γ = . If you dene rr0 r := min {r, r0}, then you can rewrite (1) as ≶ max s  2 0 r< r< kx − x k = r> 1 + − 2 cos γ, r> r> r or with ρ := < , and x := cos γ we have r> 0 p 2 kx − x k = r> 1 + ρ − 2ρx, where ρ ∈ (0, 1). The other option is: √ √ 1  1 1 + ρ2 kx − x0k = 2rr0 z − x, where z = ρ + = ∈ (1, ∞) 2 ρ 2ρ

Howard Cohl (NIST) Generalized expansions April 15, 2014 22 / 56 Generalized expansions Szeg® transformation

1 + ρ2 z = f(ρ) = 2ρ

f

Howard Cohl (NIST) Generalized expansions April 15, 2014 23 / 56 Generalized expansions Expansion of an analytic function in orthogonal polynomials

Theorem (Szeg®): Let the weight function w(x) on [−1, 1] have a geometric mean, and let {pn(x)} be the associated orthonormal systems of polynomials. Let f be an analytic function on [−1, 1] and let ∞ X f(x) = fnpn(x) (2) n=0 and Z 1 fn = f(x)pn(x)w(x)dx −1 be its orthogonal polynomial expansion. Let E be the largest ellipse with foci at ±1 in the interior of which f is regular. Then the orthogonal polynomial expansion (2) is convergent with the sum f(x) in the interior of E and divergent in the exterior of E. The convergence is uniform on every closed set lying in the interior of the ellipse.

Howard Cohl (NIST) Generalized expansions April 15, 2014 24 / 56 Generalized expansions

Howard Cohl (NIST) Generalized expansions April 15, 2014 25 / 56 Generalized expansions Obtaining generalizations: Gegenbauer generating function

Gegenbauer generating function: ∞ 1 X = ρnCν(x) (1 + ρ2 − 2ρx)ν n n=0 Connection relation: bn/2c ν 1 X (ν − µ)k (ν)n−k µ Cn(x) = (µ + n − 2k) Cn−2k(x). µ k!(µ + 1)n−k k=0 Gegenbauer expansion:

µ+ 1 iπ(µ−ν+ 1 ) ∞ 1 2 2 Γ(µ)e 2 ν−µ− 1 X 2 µ (3) ν = √ ν−µ 1 (n + µ)Q 1 (z)Cn (x) (z − x) 2 − n+µ− 2 π Γ(ν)(z − 1) 2 4 n=0 Implies Chebyshev 1st kind expansion (see Cohl & Dominici (2011)): r 1 ∞ iπ( 2 −ν) 1 1 2 e X ν− 2 ν = ν 1 nQ 1 (z)Tn(x), (z − x) π 2 − n− 2 Γ(ν)(z − 1) 2 4 n=0 Howard Cohl (NIST) Generalized expansions April 15, 2014 26 / 56 Generalized expansions Jacobi generalizations of Gegenbauer generating function

Gegenbauer→Jacobi generating function:

∞ 1 1 1 X n (2ν)n (ν− 2 ,ν− 2 ) = ρ Pn (x) (1 + ρ2 − 2ρx)ν (ν + 1 ) n=0 2 n Jacobi connection relation (2 free parameters) Ismail (2005) n (α,β) X (γ,δ) Pn (x) = cn,k(α, β; γ, δ)Pk (x), k=0

(α + k + 1) (n + α + β + 1) Γ(γ + δ + k + 1) c (α, β; γ, δ) := n−k k n,k (n − k)!Γ(γ + δ + 2k + 1) −n + k, n + k + α + β + 1, γ + k + 1 ! × 3F2 ; 1 α + k + 1, γ + δ + 2k + 2

Howard Cohl (NIST) Generalized expansions April 15, 2014 27 / 56 Generalized expansions Jacobi generalization of Gegenbauer generating function

Theorem Let α, β > −1, z ∈ C \ (−∞, 1] → C, on any ellipse with foci at ±1 and x in the interior of that ellipse. Then 1 (z − 1)α+1−ν(z + 1)β+1−ν = (z − x)ν 2α+β+1−ν ∞ X (2n + α + β + 1)Γ(α + β + n + 1)(ν)n (α+1−ν,β+1−ν) × Q (z)P (α,β)(x) Γ(α + n + 1)Γ(β + n + 1) n+ν−1 n n=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 28 / 56 Generalized expansions Jacobi function of the second kind

(α,β) Jacobi function of the 2nd kind Qγ : C \ (−∞, 1] → C dened by

2α+β+γΓ(α + γ + 1)Γ(β + γ + 1) Q(α,β)(z) := γ Γ(α + β + 2γ + 2)(z − 1)α+γ+1(z + 1)β

 γ + 1, α + γ + 1 2  × F ; , 2 1 α + β + 2γ + 2 1 − z where α + γ, β + γ∈ / −N. Lemma. Let , 1 3 5 , , n ∈ N0 µ ∈ C \ {− 2 , − 2 , − 2 ,...} ν ∈ C \ −N0 z ∈ C \ (−∞, 1], 1 1 1 1 µ−ν+ 1  iπ(µ−ν+ ) 1 (µ−ν+ ,µ−ν+ ) 2 2 Γ µ + n + e 2 ν−µ− Q 2 2 (z) = 2 Q 2 (z). n+ν−1 2 (µ−ν)/2+1/4 1 Γ(ν + n)(z − 1) n+µ− 2

Howard Cohl (NIST) Generalized expansions April 15, 2014 29 / 56 Generalized expansions Tom Koornwinder mention

The above expansion is consistent with the Jacobi binomial expansion

given in Koekoek & Koekoek (2007), namely for n ∈ N0,

(z − x)n = (−1)n2nn!Γ(α + β + 1)

n X (2k + α + β + 1)(α + β + 1)k (−α−n−1,−β−n−1) (α,β) × P (z)P (x) Γ(α + β + n + k + 2) n−k k k=0 The consistency is established through the formula

(−1)n+kΓ(α + β + n + k + 2) P (−α−n−1,−β−n−1)(z) = n−k 2α+β+2n+1(n − k)! Γ(α + k + 1)Γ(β + k + 1)

α+n+1 β+n+1 (α+n+1,β+n+1) ×(z − 1) (z + 1) Qk−n−1 (z)

Howard Cohl (NIST) Generalized expansions April 15, 2014 30 / 56 Generalized expansions (α,β) Jacobi polynomials: Pn : C → C DLMF (18.12.3) generating function 1  α+β+1 , α+β+2 2ρ(1 + x) F 2 2 ; (1 + ρ)α+β+1 2 1 β + 1 (1 + ρ)2  2 β/2 Γ(β + 1) 1 + ρ = P −β ρ(1 + x) Rα+1 α R ∞ X (α + β + 1)n = ρnP (α,β)(x), (β + 1) n n=0 n Ismail (2005) (4.3.2) generating function (α + β + 1)(1 + ρ)  α+β+2 , α+β+3 −2ρ(1 − x) F 2 2 ; (1 − ρ)α+β+2 2 1 α + 1 (1 − ρ)2  2 α/2 (α + β + 1)(1 + ρ)Γ(α + 1) 1 − ρ = P−α ρ(1 − x) Rβ+2 β+1 R ∞ X (α + β + 1)n = (2n + α + β + 1) ρnP (α,β)(x), (α + 1) n n=0 n Howard Cohl (NIST) Generalized expansions April 15, 2014 31 / 56 Generalized expansions Generalized expansions for Jacobi polynomials

Theorem

Let m ∈ N0, α, β > −1, x ∈ [−1, 1], ρ ∈ {z ∈ C : |z| < 1}\ (−1, 0]. Then

(1 + x)−β/2 1 + ρ P −β Rα+m+1 α+m R

−(α+1)/2 ∞ ρ X (2n + α + β + 1)Γ(α + β + n + 1)(α + β + m + 1)2n = 2β/2(1 − ρ)m Γ(β + n + 1) n=0 1 + ρ ×P −α−β−2n−1 P (α,β)(x). −m 1 − ρ n

Howard Cohl (NIST) Generalized expansions April 15, 2014 32 / 56 Generalized expansions Generalized expansions for Jacobi polynomials

Theorem Let α ∈ C, γ, β > −1, ρ ∈ {z ∈ C : |z| < 1}\ (−1, 0], x ∈ [−1, 1]. Then

−β/2 ! (1 + x) −β 1 + ρ Pα (1 + ρ2 − 2ρx)(α+1)/2 p1 + ρ2 − 2ρx

Γ(γ + β + 1) = 2β/2Γ(β + 1)(1 − ρ)α−γρ(γ+1)/2

∞ X (2k + γ + β + 1)(γ + β + 1)k (α + β + 1)2k × (β + 1)k k=0 1 + ρ ×P −γ−β−2k−1 P (γ,β)(x). γ−α 1 − ρ k

Howard Cohl (NIST) Generalized expansions April 15, 2014 33 / 56 Generalized expansions

−β/2 ! (1 + x) −β 1 + ρ Pα (1 + ρ2 − 2ρx)(α+1)/2 p1 + ρ2 − 2ρx ∞   Γ(γ + β + 1) X (2n + γ + β + 1)(γ + β + 1)n(α + β + 1)2n 1 + ρ = P −γ−β−2n−1 P (γ,β)(x) 2β/2Γ(β + 1)(1 − ρ)α−γρ(γ+1)/2 (β + 1) γ−α 1 − ρ n n=0 n Theorem 1 in Cohl & MacKenzie (2013) JCA and Theorem 4 in Cohl, MacKenzie & Volkmer (2013) JMAA

∞ ∞ √ ∞ 1 X 1 X 1 2 X = (2n + 1)Q (z)P (x) = ρnCν(x) √ =  Q (z)T (x) z − x n n (1 + ρ2 − 2ρx)ν n z − x π n n−1/2 n n=0 n=0 n=0 Heine (1851) Heine's formula Gegenbauer (1874) generating function Heine (1881)

∞ r 1 (z2 − 1)(1−ν)/2 X 1 2 (z2 − 1)−ν/2+1/4 = (2n + 1)Qν−1(z)P (x) = (z − x)ν eiπ(ν−1)Γ(ν) n n (z − x)ν π eiπ(ν−1/2)Γ(ν) n=0 ∞ (13) in Cohl (2013) ITSF X ν−1/2 × nQn−1/2(z)Tn(x) n=0 (3.10) in Cohl & Dominici (2011) PRA µ+1/2 iπ(µ−1/2) ∞ 1 2 Γ(µ)e X −µ+1/2 = √ (n + µ)Q (z)Cµ(x) z − x π(z2 − 1)−µ/2+1/4 n+µ−1/2 n n=0 (7.2) in Durand, Fishbane & Simmons (1976) 1 2µ+1/2Γ(µ)eiπ(µ−ν+1/2) = √ (z − x)ν π Γ(ν)(z2 − 1)(ν−µ)/2−1/4 ∞ X ν−µ−1/2 µ α β ∞ × (n + µ)Q (z)C (x) 1 (z − 1) (z + 1) X (2n + α + β + 1)Γ(α + β + n + 1)n! n+µ−1/2 n = Q(α,β)(z)P (α,β)(x) z − x 2α+β Γ(α + 1 + n)Γ(β + 1 + n) n n n=0 n=0 Theorem 2.1 in Cohl (2013) ITSF (9.2.1) in Szeg® (1959)

α+1−ν β+1−ν ∞ 1 (z − 1) (z + 1) X (2n + α + β + 1)Γ(α + β + n + 1)(ν)n (α+1−ν,β+1−ν) = Q (z)P (α,β)(x) (z − x)ν 2α+β+1−ν Γ(α + n + 1)Γ(β + n + 1) n+ν−1 n n=0 Theorem 1 in Cohl (2013) SIGMA

Howard Cohl (NIST) Generalized expansions April 15, 2014 34 / 56 Expansions for Jacobi polynomials Other Jacobi generating functions

Connection relation (1 free parameter)

(α,β) (β + 1)n Pn (x) = (γ + β + 1)(γ + β + 2)n n X (γ + β + 2k + 1)(γ + β + 1)k (n + β + α + 1)k(α − γ)n−k (γ,β) × Pk (x). (β + 1)k (n + γ + β + 2)k(n − k)! k=0

Generating function for Jacobi polynomials: DLMF (18.12.1)

α+β ∞ 2 X n (α,β) α β = ρ Pn (x), R (1 + R − ρ) (1 + R + ρ) n=0

where R := p1 + ρ2 − 2ρx.

Howard Cohl (NIST) Generalized expansions April 15, 2014 35 / 56 Expansions for Jacobi polynomials Generalized expansions for Jacobi polynomials

Theorem Let α ∈ C, γ, β > −1, ρ ∈ {z ∈ C : |z| < 1}, x ∈ [−1, 1]. Then 2α+β R (1 + R − ρ)α (1 + R + ρ)β     ∞ (2k + γ + β + 1)(γ + β + 1) α+β+1 α+β+2 1 X k 2 2 = k k γ + β + 1  γ+β+2   γ+β+3  k=0 (α + β + 1)k 2 k 2 k ! β + k + 1, α + β + 2k + 1, α − γ k (γ,β) × 3F2 ; ρ ρ P (x). α + β + k + 1, γ + β + 2k + 2 k

Howard Cohl (NIST) Generalized expansions April 15, 2014 36 / 56 Expansions for Jacobi polynomials Generalized expansions for Jacobi polynomials

Theorem Let α ∈ C, γ, β > −1, ρ ∈ {z ∈ C : |z| < 1}, x ∈ [−1, 1]. Then  2 α/2  2 β/2     J p2(1 − x)ρ I p2(1 + x)ρ (1 − x)ρ (1 + x)ρ α β 1 = (γ + β + 1)Γ(α + 1)Γ(β + 1)     ∞ (2k + γ + β + 1)(γ + β + 1) α+β+1 α+β+2 X k 2 2 × k k  γ+β+2   γ+β+3  k=0 (α + 1)k (β + 1)k (α + β + 1)k 2 k 2 k   × F 2k + α + β + 1, α − γ ; ρ ρkP (γ,β)(x) 2 3 α + β + k + 1, γ + β + 2k + 2, α + k + 1 k ∞ X ρn = P (α,β)(x). Γ(α + 1 + n)Γ(β + 1 + n) n n=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 37 / 56 Expansions for Jacobi polynomials Gegenbauer polynomials: µ Cn : C → C

Generating function from (9.8.32) of Koekoek, Lesky & Swarttouw (2010) ! ! λ, 2α − λ 1 − R − ρ λ, 2α − λ 1 − R + ρ F ; F ; 2 1 1 2 2 1 1 2 α + 2 α + 2 Γ2(α + 1 ) = 2 (1 − x2)1/4−α/2ρ1/2−α 21/2−α 1/2−α p 2  1/2−α p 2  × Pα−λ−1/2 1 + ρ − 2ρx + ρ Pα−λ−1/2 1 + ρ − 2ρx − ρ

∞ X (λ)n (2α − λ)n = ρnCα(x). (2α) (α + 1 ) n n=0 n 2 n

Howard Cohl (NIST) Generalized expansions April 15, 2014 38 / 56 Expansions for Jacobi polynomials Generalized expansions for Gegenbauer polynomials

Theorem Let , 1 . α, µ ∈ C ν ∈ (− 2 , ∞) \{0}, ρ ∈ {z ∈ C : |z| < 1} , x ∈ [−1, 1] Then (1 − x2)1/4−µ/2

1/2−µ p 2  1/2−µ p 2  ×Pµ−α−1/2 1 + ρ − 2ρx + ρ Pµ−α−1/2 1 + ρ − 2ρx − ρ

1/2−µ ∞ 2 X (α)n (2µ − α)n (µ)n = ρµ+n−1/2 Γ(µ + 1/2) (2µ) (ν) Γ(µ + n + 1/2) n=0 n n   α+n , α+n+1 , 2µ−α+n , 2µ−α+n+1 , µ − ν, µ + n 2 2 2 2 2 ν × 6F5  1 3 ; ρ  Cn(x). 2µ+n 2µ+n+1 µ+n+ 2 µ+n+ 2 2 , 2 , 2 , 2 , ν + 1 + n

Howard Cohl (NIST) Generalized expansions April 15, 2014 39 / 56 Expansions for Jacobi polynomials Associated Legendre function, Ferrers functions, and Gegenbauer polynomials

If n ∈ N0, then through DLMF (14.3.22)

2µ−1/2Γ(µ)n! P 1/2−µ (z) = √ (z2 − 1)µ/2−1/4 Cµ(z), n+µ−1/2 π Γ(2µ + n) n and from DLMF (14.3.21), one has

2µ−1/2Γ(µ)n! P1/2−µ (x) = √ (1 − x2)µ/2−1/4 Cµ(x). n+µ−1/2 π Γ(2µ + n) n

Howard Cohl (NIST) Generalized expansions April 15, 2014 40 / 56 Expansions for Jacobi polynomials Gegenbauer expansions

For specic values can be this nice expansion

2 m (2µ) X (−m)n(2µ + m)n Cµ (R + ρ)Cµ (R − ρ) = m ρnCµ(x), m m (m!)2 (2µ) (µ + 1 ) n n=0 n 2 n and from the above generalized result we have

2 m (2µ) X (ν + n)(−m)n (2µ + m)n (µ)n Cµ (R + ρ)Cµ (R − ρ) = m m m ν(m!)2 (2µ) (µ + 1/2) (ν + 1) n=0 n n n   −m+n , −m+n+1 , 2µ+m+n , 2µ+m+n+1 , µ − ν, µ + n 2 2 2 2 2 n ν × 6F5  1 3 ; ρ  ρ Cn(x), 2µ+n 2µ+n+1 µ+n+ 2 µ+n+ 2 2 , 2 , 2 , 2 , ν + 1 + n which reduces to above formula when ν = µ.

Howard Cohl (NIST) Generalized expansions April 15, 2014 41 / 56 Expansions in Wilson polynomials Wilson polynomials

2 Wn(x ; a, b, c, d) := (a + b)n(a + c)n(a + d)n −n, n + a + b + c + d − 1, a + ix, a − ix ! × 4F3 ; 1 . a + b, a + c, a + d

Connection relation with one free parameter for the Wilson polynomials

n X n! W x2; a, b, c, d = W x2; a, b, c, h n k!(n − k)! k k=0

(n + a + b + c + d − 1)k (d − h)n−k (k + a + b)n−k (k + a + c)n−k × −1 . (k + b + c)n−k(k + a + b + c + h − 1)k (2k + a + b + c + h)n−k

Howard Cohl (NIST) Generalized expansions April 15, 2014 42 / 56 Expansions in Wilson polynomials Generalized generating function for Wilson polynomials

Theorem Let ρ ∈ {z ∈ C : |z| < 1} , x ∈ (0, ∞), < a, < b, < c, < d, < h > 0 and non-real parameters a, b, c, d, h occurring in conjugate pairs. Then ! ! a + ix, b + ix c − ix, d − ix 2F1 ; ρ 2F1 ; ρ a + b c + d

∞ X (k + a + b + c + d − 1)k = (k + a + b + c + h − 1)k (a + b)k (c + d)k k! k=0 ! d − h, 2k + a + b + c + d − 1, k + a + c, k + b + c × 4F3 ; ρ k + a + b + c + d − 1, 2k + a + b + c + h, k + c + d

k 2  × ρ Wk x ; a, b, c, h .

Howard Cohl (NIST) Generalized expansions April 15, 2014 43 / 56 Expansions in Wilson polynomials Generalized generating function for Wilson polynomials

Theorem Let ρ ∈ C, |ρ| < 1, x ∈ (0, ∞), and a, b, c, d, h complex parameters with positive real parts, non-real parameters occurring in conjugate pairs among a, b, c, d and a, b, c, h. Then

(1 − ρ)1−a−b−c−d  1 (a + b + c + d − 1), 1 (a + b + c + d), a + ix, a − ix 4ρ  × F 2 2 ; − 4 3 a + b, a + c, a + d (1 − ρ)2

∞ X (k + a + b + c + d − 1)k(a + b + c + d − 1)k = ρk (k + a + b + c + h − 1)k(a + b)k(a + c)k(a + d)kk! k=0 2k + a + b + c + d − 1, d − h, k + b + c  × F ; ρ W (x2; a, b, c, h) 3 2 2k + a + b + c + h, a + d + k k

Howard Cohl (NIST) Generalized expansions April 15, 2014 44 / 56 Expansions in continuous dual Hahn polynomials Continuous dual Hahn polynomials

−n, a + ix, a − ix  S (x2; a, b, c) := (a + b) (a + c) F ; 1 , n n n 3 2 a + b, a + c where a, b, c > 0, except for possibly a pair of complex conjugates with positive real parts. Connection coecient for the continuous dual Hahn polynomials with two free parameters. Lemma Let x ∈ (0, ∞), and a, b, c, f, g ∈ C with positive real parts and non-real values appearing in conjugate pairs among a, b, c and a, f, g. Then

n X n! S (x2; a, b, c)= (k + a + b) (k + a + c) S (x2; a, f, g) n k!(n − k)! n−k n−k k k=0 k − n, k + a + f, k + a + g  × F ; 1 . 3 2 k + a + b, k + a + c

Howard Cohl (NIST) Generalized expansions April 15, 2014 45 / 56 Expansions in continuous dual Hahn polynomials Generalized gen. fn. for continuous dual-Hahn polynomials

Theorem Let ρ ∈ C with |ρ| < 1, x ∈ (0, ∞) and a, b, c, d, f > 0 except for possibly pairs of complex conjugates with positive real parts among a, b, c and a, d, f. Then a + ix, b + ix  (1 − ρ)−d+ix F ; ρ 2 1 a + b ∞ 2 k   X Sk(x ; a, d, f)ρ b − f, k + a + d = 2F1 ; ρ . (a + b)kk! k + a + b k=0 Theorem Let ρ ∈ C, x ∈ (0, ∞), and a, b, c, d > 0 except for possibly a pair of complex conjugates with positive real parts among a, b, c and a, b, d. Then

  ∞ k 2   ρ a + ix, a − ix X ρ Sk(x ; a, b, d) c − d e 2F2 ; −ρ = 1F1 ; ρ . a + b, a + c (a + b)k(a + c)kk! k + a + c k=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 46 / 56 Expansions in Meixner-Pollaczek Connection relation for Meixner-Pollaczek polynomials

If λ > 0 and φ ∈ (0, π), then the Meixner-Pollaczek polynomials are orthogonal

(2λ) −n, λ + ix  P (λ)(x; φ) := n einφ F ; 1 − e−2iφ . n n! 2 1 2λ

Connection relation with one free parameter:

Lemma Let λ > 0, φ, ψ ∈ (0, π). Then

n 1 X (2λ + k)n−k (λ) P (λ)(x; φ) = sink φ sinn−k(ψ − φ)P (x; ψ). n sinn ψ (n − k)! k k=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 47 / 56 Expansions in Meixner-Pollaczek Expansions for Meixner-Pollaczek polynomials

Theorem Let λ > 0, ψ, φ ∈ (0, π), x ∈ R, and ρ ∈ C such that

|ρ|(sin φ + | sin(ψ − φ)|) < sin ψ. Then

(1 − eiφρ)−λ+ix(1 − e−iφρ)−λ−ix  −2λ ∞ sin(ψ − φ) X (λ) = 1 − ρ P (x; ψ)˜ρk, sin ψ k k=0 where ρ sin φ ρ˜ = . sin ψ − ρ sin(ψ − φ)

etc. for continuous Hahn polynomials

Howard Cohl (NIST) Generalized expansions April 15, 2014 48 / 56 Expansions in Rogers polynomials Some results for q-Pochhammer symbols

n (a; q)n+k = (a; q)n(aq ; q)k, 2 (a; q)2n = (a, aq; q )n,

−n −1 n −n−(n) (aq ; q)n = (a q; q)n(−a) q 2 , a 6= 0,

n (a; q)k k (aq ; q)k = (aq ; q)n, (a; q)n 2 2 (a ; q )n = (a, −a; q)n,

2 2 (−a ; q )n = (ia, −ia; q)n.

Howard Cohl (NIST) Generalized expansions April 15, 2014 49 / 56 Expansions in Rogers polynomials The q-ultraspherical/Rogers polynomials

Denition (orthogonal) for |β| < 1

2 −n/2  −n 2 n 1/2 iθ 1/2 −iθ  (β ; q)nβ q , β q , β e , β e Cn(x; β|q) := 4φ3 1/2 1/2 ; q, q (q; q)n βq , −β, −βq

2 −n −inθ −n 2iθ ! (β ; q)nβ e q , β, βe = 3φ2 ; q, q (q; q)n β2, 0

inθ q−n, β ! (β; q)ne −1 −2iθ = 2φ1 ; q, qβ e , (q; q)n β−1q1−n

where x = cos θ.

Howard Cohl (NIST) Generalized expansions April 15, 2014 50 / 56 Expansions in Rogers polynomials

Theorem Let x ∈ [−1, 1], β ∈ (−1, 1) \{0}, q ∈ (0, 1). Then

1 iθ 1 iθ ! 1 −iθ 1 1 −iθ ! β 2 e , (βq) 2 e −iθ −β 2 e , −β 2 q 2 e iθ 2φ1 1 ; q, e t 2φ1 1 ; q, e t βq 2 βq 2

∞ 1 n n 1 X (β, −β, −βq 2 ; q)n(1 − γq )t = Cn(x; γ|q) 1 − γ 2 1 n=0 (β , βq 2 , qγ; q)n

n n 1 n 1 n+1 1 n+1 1 β/γ, βq , i(βq ) 2 , −i(βq ) 2 , i(βq ) 2 , −i(βq ) 2 , × φ 10 9 n+1 n/2 n/2 (n+1)/2 (n+1)/2 n+ 1 1 γq , βq , −βq , βq , −βq , (βq 2 ) 2 , n+ 1 1 n+ 1 1 n+3/2 1 n+3/2 1 ! i(βq 2 ) 2 , −i(βq 2 ) 2 , i(βq ) 2 , −i(βq ) 2 ; q, γt2 n+ 1 1 n+3/2 1 n+3/2 1 −(βq 2 ) 2 , (βq ) 2 , −(βq ) 2 ∞ 1/2 X (−β, −βq ; q)n = C (x; β|q)tn. (β2, βq1/2; q) n n=0 n

Howard Cohl (NIST) Generalized expansions April 15, 2014 51 / 56 Expansions in Rogers polynomials

Theorem Let x ∈ [−1, 1], β ∈ (−1, 1) \{0}, q ∈ (0, 1). Then

iθ  2iθ  (γe t; q)∞ γ, β, βe −iθ iθ 3φ2 2 iθ ; q, e t (e t; q)∞ β , γe t

∞ n n 1 X (β, γ; q)n(1 − γq )t = C (x; γ|q) 1 − γ (β2, qγ; q) n n=0 n

n n 1 n 1 n+1 1 n+1 1 ! β/γ, βq , (γq ) 2 , −(γq ) 2 , (γq ) 2 , −(γq ) 2 × φ ; q, γt2 6 5 βqn/2, −βqn/2, βq(n+1)/2, −βq(n+1)/2, γqn+1

∞ X (γ; q)n = C (x; β|q)tn, (β2; q) n n=0 n etc. for q-ultraspherical/Rogers polynomials.

Howard Cohl (NIST) Generalized expansions April 15, 2014 52 / 56 Expansions in q-Laguerre polynomials The q-Laguerre polynomials

Denition (orthogonal) α > −1

α+1  −n  (α) (q ; q)n q n+α+1 Ln (x; q) = 1φ1 α+1 ; q, −q x (q; q)n q  −n  1 q , −x n+α+1 = 2φ1 ; q, q . (q; q)n 0 Connection relation

(α) Ln (x; q) n(α−β) n q X n−j n−j n−j+1 j−n+β−α+1 (β) = (−1) q( 2 )(q ; q) (q ; q) L (x; q) (q; q) j n−j j n j=0

Howard Cohl (NIST) Generalized expansions April 15, 2014 53 / 56 Expansions in q-Laguerre polynomials Generalized generating function for q-Laguerre

Theorem Let α > −1. Then   1 − α+1 0φ1 α+1 ; q, −xtq (t; q)∞ q

∞ (α−β) n (β)  α−β  X (q t) Ln (x; q) q , 0 = φ ; q, t , (qα+1; q) 2 1 qα+n+1 n=0 n

∞ (α) X Ln (x; q) = tn. (qα+1; q) n=0 n

Howard Cohl (NIST) Generalized expansions April 15, 2014 54 / 56 Expansions in q-Laguerre polynomials Generalized generating function for q-Laguerre (cont.)

Theorem Let α > −1. Then

 −  (t; q) φ ; q, −qα+1xt ∞ 0 2 qα+1, t

n ∞ α−β n ( ) (β)  α−β  X (−tq ) q 2 Ln (x; q) q = φ ; q, qnt (qα+1; q) 1 1 qα+n+1 n=0 n

∞ n (n) X (−1) q 2 = L(α)(x; q)tn. (qα+1; q) n n=0 n

etc. for q-Laguerre and etc. for little q-Laguerre.

Howard Cohl (NIST) Generalized expansions April 15, 2014 55 / 56 Final Remarks Final Remarks & see publications

Denite integrals from orthogonality for the basic case Discrete hypergeometric orthogonal polynomials Big/little q-Jacobi, continuous q-Jacobi, Askey-Wilson On a generalization of the generating function for Gegenbauer polynomials, H. S. Cohl, 2013, Integral Transforms and Special Functions, 24, 10, 807816, 10 pp. Generalizations and specializations of generating functions for Jacobi, Gegenbauer, Chebyshev and Legendre polynomials with denite integrals, H. S. Cohl and Connor MacKenzie, 2013, Journal of Classical Analysis, 3, 1, 17-33, 17 pp. Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems, H. S. Cohl, 2013, Symmetry, Integrability and Geometry: Methods and Applications, 9, 042, 26 pp. Generalizations of generating functions for hypergeometric orthogonal polynomials with denite integrals, H. S. Cohl, Connor MacKenzie, and H. Volkmer, 2013, Journal of Mathematical Analysis and Applications, 407, 2, 211-225, 15 pp. Generalized generating functions for higher continuous orthogonal polynomials in the Askey scheme, M. A. Baeder, H. S. Cohl, H. Volkmer (submitted).

Howard Cohl (NIST) Generalized expansions April 15, 2014 56 / 56