Taiwanese Journal of Mathematics

Notes On Carlitz's q-Operators

Jian Cao

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In this paper, Carlitz’s $q$-operator and the auxiliary ones are applied to prove $q$-Christoffel-Darboux formulas and some Carlitz type generating functions. In addition, the technique of exponential operator decomposition to deduce $q$-Mehler’s formulas for Rogers-Szego and Hahn polynomials are shown.

Article information

Taiwanese J. Math., Volume 14, Number 6 (2010), 2229-2244.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 05A30: $q$-calculus and related topics [See also 33Dxx] 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] 33D15: Basic hypergeometric functions in one variable, $_r\phi_s$ 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Carlitz's $q$-operators Rogers-Szegö polynomials $q$-Mehler's formula Hahn polynomials $q$-Christoffel-Darboux formulas exponential operator decomposition


Cao, Jian. Notes On Carlitz's q-Operators. Taiwanese J. Math. 14 (2010), no. 6, 2229--2244. doi:10.11650/twjm/1500406072. https://projecteuclid.org/euclid.twjm/1500406072

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  • W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. J., 31 (1964), 127-142.
  • W. A. Al-Salam and M. E. H. Ismail, $q$-beta integrals and the $q$-hermite polynomials, Pacific J. Math., 135 (1988), 209-221.
  • W. A. Al-Salam and L. Carlitz, Some orthogonal $q$-polynomials, Math. Nachr., 30 (1965), 47-61.
  • D. Bowman, $q$-difference operator, orthogonal polynomials, and symmetric expansions, Mem. Amer. Soc., 159 (2002).
  • J. L. Burchnall, A note on the polynomials of Hermite, Quart. J. Math. Oxford Ser., 12 (1941), 9-11.
  • J. Cao, New proofs of generating functions for Rogers-Szegö polynomials, Appl. Math. Comp., 207 (2009), 486-492.
  • L. Carlitz, Some polynomials related to theta functions, Annali di Matematica Pura ed Applicata, 41 (1956), 359-373.
  • L. Carlitz, Some polynomials related to theta functions, Duke Math. Jour., 24 (1957), 521-527.
  • L. Carlitz, A note on the Laguerre polynomials, Michigan Math. Jour., 7 (1960), 219-223.
  • L. Carlitz, A $q$-identity, Monatsh. für. Math., 67 (1963), 305-310.
  • L. Carlitz, Generating functions for certain $q$-orthogonal polynomials, Collectanea Math., 23 (1972), 91-104.
  • W. Y. C. Chen and Z.-G. Liu, Parameter augmentation for basic hypergeometric series II, J. Combin. Theory Ser. A, 80 (1997), 175-195.
  • W. Y. C. Chen and Z.-G. Liu, Parameter augmentation for basic hypergeometric series, I, in: Mathematical Essays in honor of Gian-Carlo Rota, B. E. Sagan and R. P. Stanley (Eds.), Birkäuser, Basel, 1998, pp. 111-129.
  • A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions II, McGraw-Hill, New York, (1953).
  • G. Dattoli, A. Torre and M. Carpanese, Operational rules and arbitrary order Hermite generating functions, J. Math. Anal. Appl., 227 (1998), 98-111.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, MA, 1990.
  • H. W. Gould and A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29 (1962), 51-64.
  • M. E. H. Ismail, D. Stanton and G. Viennot, The combinatorics of $q$-Hermite polynomials and the Askey-Wilson integral, Europ. J. Combinatorics, 8 (1987), 379-392.
  • Z.-G. Liu, $q$-Hermite polynomials and a $q$-Beta integral, Northeast Math. J., 13 (1997), 361-366.
  • Z.-G. Liu, Some identiites of differential operator and Hermite polynomials, J. Math. Res. Exp., 18 (1998), 412-416, (in Chinese).
  • N. Nielsen, Recherches sur les polynomes d'Hermite, Kgl. Danske Vidensk. Selskat. mat.-fys. Meddelelser (I), 6 (1918), 1-78.
  • L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc., 24 (1893), 337-352.
  • G. Szegö, Ein Beitrag zur Theorie der Thetafunktionen, Sitz. Preuss. Akad. Wiss., Phys. Math. Klasse, 19 (1926), 242-252.
  • H. M. Srivastava and V. K. Jain, Some multilinear generating functions for $q$-hermite polynomials, J. Math. Anal. Appl., 144 (1989), 147-157.
  • H. M. Srivastava and H. L. manocha, A treatise on generating functions, Ellis Horwood, New York, 1984.