q-Calculus as operational algebra

Abstract

This second paper on operational calculus is a continuation of Ernst, T. q-Analogues of some operational formulas. Algebras Groups Geom., 2006, 23(4), 354-374. We find multiple q-analogues of formulas in Carlitz, L. A note on the Laguerre polynomials. Michigan Math. J., 1960, 7, 219-223, for the Cigler q-Laguerre polynomials (Ernst, T. A method for q-calculus. J. Nonlinear Math. Phys., 2003, 10(4), 487-525). The q-Jacobi polynomials (Jacobi, C. G. J. Werke 6. Berlin, 1891) are treated in the same way, we generalize further to q-analogues of Manocha, H. L. and Sharma, B. L. (Some formulae for Jacobi polynomials. Proc. Cambridge Philos. Soc., 1966, 62, 459-462) and Singh, R. P. (Operational formulae for Jacobi and other polynomials. Rend. Sem. Mat. Univ. Padova, 1965, 35, 237-244). A field of fractions for Cigler's multiplication operator (Cigler, J. Operatormethoden für q-Identitäten II, q-Laguerre-Polynome. Monatsh. Math., 1981, 91, 105-117) is used in the computations. The formulas for q-Jacobi polynomials are mostly formal. We find q-orthogonality relations for q-Laguerre, q-Jacobi, and q-Legendre polynomials using q-integration by parts. This q-Legendre polynomial is given here for the first time, we also find its q-difference equations. An inequality for a q-exponential function is proved. The q-difference equation for pφp-1 (a1,...,ap;b 1,... ,bp-1 |q,z) is given improving on Smith, E. R. Zur Theorie der Heineschen Reihe und Hirer Verallgemeinerung. Diss. Univ. Munchen 1911, p. 11, by using ek =elementary symmetric polynomial. Partial q-difference equations for the q-Appell and q-Lauricella functions are found, improving on Jackson, F. H. On basic double hypergeometric functions. Quart. J. Math., Oxford Ser., 1942, 13, 69-82, and Gasper, G. and Rahman, M. Basic hypergeometric series. Second edition. Cambridge, 2004, p. 299, where q-difference equations for q-Appell functions were given with different notation. The q-difference equation for φ1 can also be written in canonical form, a q-analogue of [p. 146] Mellin, H. J. Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichunge, Acta Math., 1901, 25, 139-164.