The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

摘要

In the introduction, the main object of the paper, namely, the calculation and study of the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the Jacoby and the Gegenbauer polynomials, is briefly discussed. Section 2 contains the underlying theory restricted to the citation of a few general formulas regarding the orthogonality property of associated polynomials of the second kind. Section 3 which deals with the application to the case of the Jacobi polynomials {P(α,β)n(z)¦nϵN}, starts with the construction of the orthogonality relation and the weight function ϱ1 of the associated polynomials for arbitrary α,β\gT-1. The denominator of the fraction representing ϱ1 comprises the principal value of an integral which is studied in detail in order to bring it in an explicit form most suited for practical use. Attention is also paid to the associated functions of the second kind and some of their properties. Next, in a subsection, ϱ1 is expressed solely in terms of elementary functions in a number of special cases involving subsets of (α,β)-pairs. When α+β=-1 (-1<α, β<0), ϱ1 and the associated polynomials are themselves of Jacobian type. The problem of finding all other (α, β)-pairs for which this property also holds is completely solved. Further, ϱ1 and the orthogonality relation are obtained and discussed in the cases α+β=0(-1<α,β<1), χ+β=1(-1<α,β<2), α+β=k(-1<α,β-1,β\gT0) and β=α-1 (α>;0,β\gT-1). In another subsection, the generating function for the associated polynomials is calculated and the analysis of the result comprises some related formulas. In the final subsection, the moments of the weight function ϱ1 with respect to 0, -1 and 1 are expressed in terms of the corresponding moments of the weight function of the Jacobi polynomials, both in finite determinantal form and by means of a recurrence relation. In Section 4 which concerns the application to the case of the Gegenbauer polynomials {Cvn(z)¦nϵN}, the advantages of a new definition of the associated polynomials of the second kind are at first expounded. Then follow the orthogonality relation and the weight function ϱ1 for arbitrary v\gT;-12. Next, ϱ1 is calculated in terms of elementary functions for all integer and half-odd integer values of v exceeding -12. In the former case, a remarkable simplification occurs in the denominator of the fraction representing ϱ1. After that follow three subsections devoted respectively to the functions of the second kind associated with the Gegenbauer polynomials, two different generating functions for the associated polynomials and related formulas, and the moments of the weight function ϱ1. The paper ends with two appendices in which certain types of sums are reduced to a one-term expression.