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<p>Inspired by the work of R.P. Boas and R.C. Buck [Polynomial Expansions of Analytic Functions, Springer, Berlin, 1964] and M. Simon and S.K. Suslov [Expansion of analytic functions in <i>q</i>-orthogonal polynomials, Ramanujan J. 19(2009), no. 3, 281–303] in expanding analytic functions using <i>q</i>-orthogonal polynomials, this paper explores whether analytic functions can be further decomposed into homogeneous <i>q</i>-polynomials through <i>q</i>-partial differential equations. Our primary focus lies in constructing <i>q</i>-partial differential equations, employing a recursive algorithm, specifically for homogeneous <i>q</i>-Laguerre polynomials and their associated applications. We establish that an analytic function can be represented as a product of homogeneous <i>q</i>-Laguerre polynomials if and only if it satisfies a particular <i>q</i>-partial differential equation. Furthermore, we derive bilinear, multilinear, mixed, and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(U(n+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-type generating functions for homogeneous <i>q</i>-Laguerre polynomials using <i>q</i>-partial differential equations. Additionally, we generalize the Andrews–Askey integral and Ramanujan’s integral, incorporating generating functions for homogeneous <i>q</i>-Laguerre polynomials, through the lens of <i>q</i>-partial differential equations.</p>

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Homogeneous q-Laguerre type polynomials expansion for analytic functions via q-partial differential equations as well as generating functions and q-integrals

  • 期刊论文

  • 发表日期：

    2025年10月21日

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  • Jian Cao

摘要

Inspired by the work of R.P. Boas and R.C. Buck [Polynomial Expansions of Analytic Functions, Springer, Berlin, 1964] and M. Simon and S.K. Suslov [Expansion of analytic functions in q-orthogonal polynomials, Ramanujan J. 19(2009), no. 3, 281–303] in expanding analytic functions using q-orthogonal polynomials, this paper explores whether analytic functions can be further decomposed into homogeneous q-polynomials through q-partial differential equations. Our primary focus lies in constructing q-partial differential equations, employing a recursive algorithm, specifically for homogeneous q-Laguerre polynomials and their associated applications. We establish that an analytic function can be represented as a product of homogeneous q-Laguerre polynomials if and only if it satisfies a particular q-partial differential equation. Furthermore, we derive bilinear, multilinear, mixed, and \(U(n+1)\) U ( n + 1 ) -type generating functions for homogeneous q-Laguerre polynomials using q-partial differential equations. Additionally, we generalize the Andrews–Askey integral and Ramanujan’s integral, incorporating generating functions for homogeneous q-Laguerre polynomials, through the lens of q-partial differential equations.