Explicit Formulas for the Associated Jacobi Polynomials and Some Applications

Abstract

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

First return probabilities of birth and death chains and associated orthogonal polynomials

“…This relation was already found by Grosjean [9] using a special structure of the corresponding weight function and by Wimp [22] [formula (40)] as a special case of a representation for the c-th associated Jacobi polynomial in terms of generalized hypergeometric functions.…”

The Associated Classical Orthogonal Polynomials

“…Magnus did not attempt to obtain the measure of orthogonality for the associated Askey-Wilson polynomials, p α n (x; a, b, c, d|q), which satisfy the 3-term recurrence relation (2.13) with A n , B n , C n replaced by A n+α , B n+α , C n+α , but gave a scheme of how to derive the corresponding fourth order difference equation. Wimp [49], however, was able to find an explicit fourth order differential equation for the associated Jacobi polynomials which are the q → 1 limit cases of p α n (x; q 1/2 , q α+1/2 , −q β+1/2 , −q 1/2 |q). Considering the complexity of the weight function w λ (x; a, b, c) in (1.37) for the Pollaczek polynomials it would appear that guessing the measure of orthogonality for the associated OPS or to derive it from special function formulas would be a daunting task indeed.…”

Jacobi matrices generated by ratios of hypergeometric functions

“…The explicit formula for µ(a, b, c; t) can be extracted from the findings of [5]. Also, it is worth mentioning that the closed formula for the approximants to the corresponding continued fraction can be found in [25].…”