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Riccati equations and convolution formulae for functions of Rayleigh type. (English) Zbl 0953.33003
N. Kishorne [Proc. Am. Math. Soc. 14, 527-533 (1963; Zbl 0117.29904)] provided a convolution-type sum formula for finding the Rayleigh functions defined by σ n = k=1 j νk -2n (n=1,2,) (j νk being the positive zeros of the Bessel function J ν (z)), in terms of σ 1 ,,σ n-1 . On the other hand, the second author and A. Raza [J. Phys. A, Math. Gen. 31, No. 46, 9327-9330 (1998; Zbl 0937.33005)] obtained corresponding expressions for sums of reciprocal powers of zeros τ n of the more general function N ν (z)=az 2 J ν '' (z)+bzJ ν ' (z)+cJ ν (z), in terms of τ 1 ,τ 2 ,,τ n-1 and σ 1 ,σ 2 ,,σ n . In the present paper the authors, by using certain Riccati equation satisfied by z -ν/2 N ν (z 1/2 ), deduce an expression for τ n in terms of τ k , k=1,,n-1 only. Other results for zeros of confluent hypergeometric functions which extend corresponding ones due to H. Buchholz [Z. Angew. Math. Mech. 31, 149-152 (1951; Zbl 0042.07803)] are also obtained.
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1
34B30Special ODE (Mathieu, Hill, Bessel, etc.)