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Zbl 0604.44004
Valent, G.
An integral transform involving Heun functions and a related eigenvalue problem. (English)
[J] SIAM J. Math. Anal. 17, 688-703 (1986). ISSN 0036-1410; ISSN 1095-7154/e

An integral transform involving Heun functions is obtained. When combined with the explicit solutions given by Carlitz, new closed integral representations are obtained for some Heun functions. Using this transformation, the author is able to solve an eigenvalue problem $$ \{(1-x)d/dx[x(1-k\sp 2x)d/dx]-k\sp 2/4(1-x)-s\sb 0\}y\sb 0(x)=0 $$ with $k\sp 2\in D\sb 0=\{k\sp 2\vert 0\le k\sp 2\le k\sp 2\sb 0<1\}$, $y\sb{s\sb 0}(0)=1$, $y\sb{s\sb 0}(1)=0$; and related to a birth and death process, obtaining the exact spectrum and eigenfunctions. The integral representations obtained are sufficient to give a direct proof of their orthogonality, to allow the computation of their norm and to prove their completeness in the Hilbert space $L\sp 2\sb w$.
[R.S.Dahiya]

MSC 2000:: *44A15 Special transforms
34L99 Ordinary differential operators
34A25 Analytical theory of ODE
60J80 Branching processes

Keywords: Heun's differential equation; Heun functions; eigenvalue problem; birth and death process; spectrum; integral representations

Cited in: Zbl 1134.30301

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