General recurrence and ladder relations of hypergeometric-type functions. (English) Zbl 1124.33017
The authors consider sequences of solutions \(y_{\nu_i}(z,\bar{c}_i),i=0,\ldots,M\) of the hypergeometric-type differential equation
\[
\sigma(z)y''+\tau(z;\nu,\bar{c})y'+\lambda_{\nu}(\bar{c})y=0,
\]
where \(\sigma\) and \(\tau\) are polynomials of degree not greater than two and one, respectively, \(\bar{c}\) stands for a set of meaningful parameters, \(\nu\) is a solution of the equation \(\lambda(\bar{c})+\nu\tau'(\bar{c})+\frac{\nu(\nu-1)}{2}\sigma''=0.\) Under several restrictions on \(\{y_{\nu_i}\}\) it is proven that the sum rule
\[
\sum_{i=0}^{N-1}A_{\nu_i,k_i}(z,\bar{c}_i)y_{\nu_i}^{(k_i)}(z;\bar{c}_i)=0\quad (3\leq N\leq M+1)
\]
holds with rational functions \(A_{\nu_i,k_i}(z).\)
Reviewer: Alexei Lukashov (Istanbul)
MSC:
| 33C99 | Hypergeometric functions |
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
Keywords:
second-order differential equations; special functions; hypergeometric-type functions; ladder operators; sum rulesReferences:
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