The inverse and derivative connecting problems for some Hypergeometric polynomials
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 2
Abstract
Given two polynomial sets {Pn(x)}n≥0, and {Qn(x)}n≥0 such that deg(Pn(x))=deg(Qn(x))=n. The so-called connection problem between them asks to find coefficients αn,k in the expression Qn(x)=n∑k=0αn,kPk(x). The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case Qn(x)=xn the connection problem is called the inversion problem associated to {Pn(x)}n≥0. The particular case Qn(x)=P′n+1(x) is called the derivative connecting problem for polynomial family {Pn(x)}n≥0. In this paper, we give a closed-form expression of the inversion and the derivative coefficients for hypergeometric polynomials of the form 2F1[−n,ab∣∣∣z],2F1[−n,n+ab∣∣∣z],2F1[−n,a±n+b∣∣∣z], where 2F1[a,bc∣∣∣z]=∞∑k=0(a)k(b)k(c)kzkk!, is the Gauss hypergeometric function and (x)n denotes the Pochhammer symbol defined by (x)n={1,n=0,x(x+1)(x+2)⋯(x+n−1),n>0. All polynomials are considered over the field of real numbers.
Authors and Affiliations
L. Bedratyuk, A. Bedratuyk
Keywords
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- DOI 10.15330/cmp.10.2.235-247
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