Al-Salam–Carlitz polynomials

Template:Distinguish In mathematics, Al-Salam–Carlitz polynomials UTemplate:Su(x;q) and VTemplate:Su(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Template:Harvs. Template:Harvs give a detailed list of their properties.

Definition

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

$U_n^{(a)}(x;q)=(-a{)}^n{q}^{n(n-1)/2}{}_2{\varphi }_1(q^{-n},x^{-1};0;q,qx/a)$

$V_n^{(a)}(x;q)=(-a{)}^n{q}^{-n(n-1)/2}{}_2{\varphi }_0(q^{-n},x;-;q,q^n/a)$

References

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Further reading

• Wang, M. (2009). $q$-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
• Askey, R., & Suslov, S. K. (1993). The $q$-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 $q$–Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.