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Generalizations of Orthogonal Polynomials in Askey Schemes Generalizations of Orthogonal polynomials 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 Askey scheme 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 fpn(x)g 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; 1) 2 Cts. dual Hahn S Γ(a+ix)Γ(b+ix)Γ(c+ix) (0; 1) n Γ(2ix) 2 Continuous Hahn pn jΓ(a + ix)Γ(b + ix)j (−∞; 1) (λ) (2φ−π)x 2 Meixner-Pollaczek Pn e jΓ(λ + ix)j (−∞; 1) (α,β) α β Jacobi Pn (1 − x) (1 + x) (−1; 1) −x α Laguerre Ln e x (0; 1) −x2 Hermite Hn e (−∞; 1) 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 1 π Γ(z) := tz−1e−tdt; Re z > 0; Γ(z)Γ(1 − z) = 0 sin(πz) Γ(n) = (n − 1)!; n 2 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 2 C Γ(a + n) (a) = ; a 62 −N = 0; −1; −2;::: n Γ(a) 0 Generalized hypergeometric series 1 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 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 1 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 1 X n (α) f(x; ρ; α) = cn(α)ρ Pn (x) n=0 Examples: Hermite polynomials 1 X 1 exp(2xρ − ρ2) = ρnH (x) n! n n=0 Gegenbauer polynomials 1 1 X = ρnCν(x) (1 + ρ2 − 2ρx)ν n n=0 Jacobi polynomials 1 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 1 xρ X (1 − ρ)−α−1 exp = ρnL(α)(x) ρ − 1 n n=0 1 (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 1 1 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.) 1 1 1 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 1 ( 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 dual Hahn polynomials 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: 1 x X qn(n−1)ρn (αρ; q) φ ; q; ρ = V (α)(x; q) 1 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: 1 x X qn(n−1)ρn (αρ; q) φ ; q; ρ = 1 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 1 k (β) x X ρ Vk (x; q) (αρ; q)1 1φ1 ; q; ρ = αρ (q; q)k k=0 1 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 1 (β) 1 n β k k(k−1) (2) k n x X ρ q Vk (x; q) X q ( α ; q)n(−αρq ) (αρ; q)1 1φ1 ; q; ρ = αρ (q; q)k (q; q)n k=0 n=0 therefore with α; β < 0, one has 1 n n(n−1) (β) x X ρ q Vn (x; q) β/α (αρ; q) φ ; q; ρ = φ ; q; αρqn 1 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 1 X = ρnCν(x) (1 + ρ2 − 2ρx)ν n n=0 Legendre generating function (1783) 1 1 X n = ρ Pn(x) p 2 1 + ρ − 2ρx n=0 Heine's reciprocal square root identity (1881) p 1 1 2 X 1 if n = 0; p = Q (z)T (x); := z − x π n n−1=2 n n 2 if n = 1; 2; 3;::: n=0 1 1 1 X 1 + ρ2 = p 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 (− 2 ; 1) n f0g;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 n (−∞; 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 n (−∞; 1] ! C p 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 p The alg.
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