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Zbl 0953.33003
Gupta, Dharma P.; Muldoon, Martin E.
Riccati equations and convolution formulae for functions of Rayleigh type. (English)
[J] J. Phys. A, Math. Gen. 33, No.7, 1363-1368 (2000). ISSN 0305-4470

{\it 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 $\sigma_n= \sum^\infty_{k=1} j^{-2n}_{\nu k} (n=1,2, \dots)$ $(j_{\nu k}$ being the positive zeros of the Bessel function $J_\nu(z))$, in terms of $\sigma_1,\dots,\sigma_{n-1}$. On the other hand, the second author and {\it 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 $\tau_n$ of the more general function $N_\nu(z)= az^2J_\nu'' (z)+bzJ_\nu'(z) +cJ_\nu(z)$, in terms of $\tau_1, \tau_2, \dots, \tau_{n-1}$ and $\sigma_1,\sigma_2, \dots, \sigma_n$. In the present paper the authors, by using certain Riccati equation satisfied by $z^{-\nu/2} N_\nu(z^{1/2})$, deduce an expression for $\tau_n$ in terms of $\tau_k$, $k=1, \dots, n-1$ only. Other results for zeros of confluent hypergeometric functions which extend corresponding ones due to {\it H. Buchholz} [Z. Angew. Math. Mech. 31, 149-152 (1951; Zbl 0042.07803)] are also obtained.
[N.Hayek (La Laguna)]

MSC 2000:: *33C10 Cylinder functions, etc.
33C15 Confluent hypergeometric functions
34B30 Special ODE

Keywords: zeros of Bessel functions; confluent hypergeometric functions; Rayleigh functions

Citations: Zbl 0117.29904; Zbl 0937.33005; Zbl 0042.07803

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