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Zbl 0587.41012
Wimp, Jet
Some explicit Padé approximants for the function $\Phi$ '/$\Phi$ and related quadrature formula involving Bessel functions. (English)
[J] SIAM J. Math. Anal. 16, 887-895 (1985). ISSN 0036-1410; ISSN 1095-7154/e

The author gives an explicit formula for the [n/n] Padé approximant for $\Phi$ '/$\Phi$ ($\Phi$ (a,c;z) the confluent hypergeometric function of the first kind) and for the error, generalizing a result by {\it W. A. Fair} (the same function but under the condition $c=2a=2\nu +1$, the Bessel function case [Math. Comput. 18, 627-634 (1964; Zbl 0123.326)]. Moreover the distribution function connected with a discrete orthogonality relation following from the recurrence relation for the Padé numerators and denominators is recovered from an explicit form of its Stieltjes transform. Finally restriction of one of the parameters in the problem leads to the Bessel function case where the discrete orthogonality gives rise to an exact quadrature formula for functions having a convergent Neumann series using Bessel functions of the first kind on [0,$\infty)$. Cumbersome details and lengthy calculations have been omitted (sometimes introducing rather large jumps in the proofs given): a well written and interesting paper.
[M.G.de Bruin]

MSC 2000:: *41A21 Pade approximation
33C05 Classical hypergeometric functions
42C05 General theory of orthogonal functions and polynomials
65D32 Quadrature formulas (numerical methods)

Keywords: Padé approximant; Stieltjes transform; Bessel function; quadrature formula; Neumann series

Citations: Zbl 0123.326

Cited in: Zbl 0697.39002

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