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 fur q-Identitaten 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 (p)phi(p-1) (a(1),...,a(p); b(1),...,b(p-1)vertical bar q, z) is given improving on Smith, E. R. Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung. Diss. Univ. Munchen 1911, p. 11, by using e(k) = 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 Phi(1) can also be written in canonical form, a q-analogue of [p. 146] Mellin, H. J. Uber den Zusammenhang zwischen den linearen Differential- und Differenzengleichunge, Acta Math., 1901, 25, 139-164.
Highlights
The aim of this paper is to present q-calculus as a truly operational subject
Operational formulas were often used with great success in the theory of classical orthogonal polynomials and Bessel functions
One example of operator is the Rodriguez operator operating on holomorphic functions, this is a generalization of the Rodriguez formula for Laguerre and Jacobi polynomials
Summary
The aim of this paper is to present q-calculus as a truly operational subject. Operational formulas were often used with great success in the theory of classical orthogonal polynomials and Bessel functions. One example of operator is the Rodriguez operator operating on holomorphic functions, this is a generalization of the Rodriguez formula for Laguerre and Jacobi polynomials. In [20] the equivalent approach (by the q-binomial theorem) to use the q-exponential function Eq to obtain formulas for q-Laguerre polynomials was used. We will prove many operational formulas in this paper, so we need a designation for the functions to operate upon. This class of function will be called holomorphic. Let Hq denote holomorphic functions C[[x]], or more generally, functions of the form
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