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Some explicit Padé approximants for the function ${\Phi }$ ’/${\Phi }$ and related quadrature formula involving Bessel functions. (English) Zbl 0587.41012
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 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,$\infty \right)$. Cumbersome details and lengthy calculations have been omitted (sometimes introducing rather large jumps in the proofs given): a well written and interesting paper.
Reviewer: M.G.de Bruin

##### MSC:
 41A21 Padé approximation 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 42C05 General theory of orthogonal functions and polynomials 65D32 Quadrature and cubature formulas (numerical methods)