The theory of difference analogues of special functions of hypergeometric type

Theq-harmonic oscillator and the Al-Salam and Carlitz polynomials

“…( For details, see [16][17][18][19]. ) The orthogonality property (2) or (4) can be proved by using standard SturmLiouville-type arguments ( cf.…”

“…The orthogonality property (2) or (4) can be proved by using standard SturmLiouville-type arguments ( cf. [16][17][18][19]). …”

Difference Fourier transforms for nonreduced root systems

“…In particular, E + (γ, ·) ∈ M + is a meromorphic eigenfunction of the Askey-Wilson second order q-difference operator, which admits an explicit series expansion in terms of the Askey-Wilson polynomials, see Proposition 6.14. On the other hand, a basis of eigenfunctions for the Askey-Wilson second order q-difference operator was given explicitly by Ismail and Rahman [21] in terms of very-well-poised 8 φ 7 basic hypergeometric series, see also Suslov [44]. Here the very-well-poised 8 φ 7 basic hypergeometric series, denoted by 8 W 7 , is defined by see [16] for details.…”

On Factorization and Solutions ofq-difference Equations Satisfied by some Classes of Orthogonal Polynomials

“…Under the boundary condition s ðxðsÞÞ wðxðsÞÞx k s 2 1 2 s¼a;b ¼ 0; ;k; the polynomials (P n (x(s))) n are orthogonal with respect to the weight function w (x(s)) [25,28] X b21 s¼a P n ðxðsÞÞP m ðxðsÞÞ wðxðsÞÞDx s 2 1 2 ¼ k n d n;m; k n -0; ;n;…”

“…Q n is related to P n and its first associated (associated of order r ¼ 1) by Ref. [28] Q n ðxðzÞÞ ¼ P n ðxðzÞÞQ 0 ðxðzÞÞ þ P ð1Þ n21 ðxðzÞÞ wðxðzÞÞ ; ð13Þ…”