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Interlacing of the zeros of contiguous hypergeometric functions. (English) Zbl 1171.33008

Interlacing properties of the zeros of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials) are explored. The main result to be applied for proving interlacing is the following statement (Lemma 1), proved in [J. Segura, SIAM J. Numer. Anal. 40, No. 1, 114–133 (2002; Zbl 1058.33020)] in a slightly different version: Let ${y}_{m}\left(x\right)$ and ${y}_{m-1}\left(x\right)$ be two nontrivial solutions of the system of first order difference-differential equations (DDEs)

${y}_{m}^{\text{'}}\left(x\right)={a}_{m}\left(x\right){y}_{m}\left(x\right)+{d}_{m}\left(x\right){y}_{m-1}\left(x\right),\phantom{\rule{2.em}{0ex}}{y}_{m-1}^{\text{'}}\left(x\right)={b}_{m}\left(x\right){y}_{m-1}\left(x\right)+{e}_{m}\left(x\right){y}_{m}\left(x\right),$

with continuous coefficients, as functions on $x$ in an interval $I$ and such that ${d}_{m}\left(x\right)$ and ${e}_{m}\left(x\right)$ do not change the sign in $I$. If one of these functions ${y}_{m}\left(x\right)$ or ${y}_{m-1}\left(x\right)$ has, at least, two zeros in $I$, then the zeros of ${y}_{m}\left(x\right)$ and ${y}_{m-1}\left(x\right)$ are interlaced and ${d}_{m}\left(x\right){e}_{m}\left(x\right)<0$ in $I$. A sequence of hypergeometric functions ${y}_{m}\left(x\right)={}_{2}F_{1}\left(a+{\epsilon }_{1}m,b+{\epsilon }_{2}m;c+{\epsilon }_{3}m;x\right)$ (${\epsilon }_{i},m\in ℤ$, $a,b,c,x\in ℝ$) satisfy first order DDEs with coefficients continuous in $\left(0,1\right)$. Using the above statement, in particular, it is proved that if ${p}_{n+1}\left(x\right)$ and ${p}_{n-1}\left(x\right)$ are two classical orthogonal polynomials (Hermite, Laguerre, Jacobi) with respect to the same weight function $w\left(x\right)$ in $\left[a,b\right]$, then the zeros of ${p}_{n+1}\left(x\right)$ and ${p}_{n-1}\left(x\right)$ are interlaced for $x>{\beta }_{n}$ and $x<{\beta }_{n}$, where ${\beta }_{n}={\int }_{a}^{b}x{p}_{n}^{2}\left(x\right)w\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx/{\int }_{a}^{b}{p}_{n}^{2}\left(x\right)w\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\in \left(a,b\right)$ (Theorem 1).

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 34K06 Linear functional-differential equations
##### Keywords:
hypergeometric functions; first order DDEs; zeros; interlacing
##### References:
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