Al-Salam–Carlitz polynomials

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In mathematics, Al-Salam–Carlitz polynomials U(a)n(x;q) and V(a)n(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14.24, 14.25) give a detailed list of their properties.


The Al-Salam–Carlitz polynomials are given in terms of basic hypergeometric functions by

[math] U_n^{(a)}(x;q) = (-a)^nq^{n(n-1)/2}{}_2\phi_1(q^{-n}, x^{-1};0;q,qx/a)[/math]
[math] V_n^{(a)}(x;q) = (-a)^nq^{-n(n-1)/2}{}_2\phi_0(q^{-n}, x;;q,q^n/a)[/math]


Further reading[edit]

  • Wang, M. (2009). [math]q[/math]-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
  • Askey, R., & Suslov, S. K. (1993). The [math]q[/math]-harmonic oscillator and the Al-Salam and Carlitz polynomials. Letters in Mathematical Physics, 29(2), 123-132.
  • Chen, W. Y., Saad, H. L., & Sun, L. H. (2010). An operator approach to the Al-Salam–Carlitz polynomials. Journal of Mathematical Physics, 51(4).
  • Kim, D. (1997). On combinatorics of Al-Salam Carlitz polynomials. European Journal of Combinatorics, 18(3), 295-302.
  • Andrews, G. E. (2000). Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. Contemporary Mathematics, 254, 45-56.
  • Baker, T. H., & Forrester, P. J. (2000). Multivariable Al–Salam & Carlitz Polynomials Associated with the Type A [math]q[/math]–Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.category:Special functions–Carlitz polynomials was the original source. Read more.