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Sur une généralisation des polynômes d’Hermite. (French) JFM 55.0799.01
Aus der allgemeinen Theorie der orthogonalen Polynome folgen leicht verschiedene Eigenschaften der Polynome $\varphi_n(x)$, definiert durch die Orthogonalitätsbedingung $$ \sum_{i=0}^{l-1} \binom{l-1}{i} p^i q^{l-1-i} \varphi_m(i) \varphi_n(i) = \varepsilon_{mn} \tag 1 $$ $$ (m, n = 0, 1, 2, \dots; \ p > 0, \ q > 0, \ p + q = 1). $$ Es gilt $$ \varphi_n(x) =\binom{l-1}{n}^{-\tfrac 12} (pq)^{-\tfrac n2} \sum_{i=0}^n (-1)^i \binom{l - x -1}{n - i}\binom{x}{i} p^{n-i} q^i, $$ woraus für $l\to\infty$, $p(l - 1) = a =$ const die Polynome $$ \text{const }\frac{x!}{a^x} \varDelta^n \left[ \frac{a^{x-n}}{(x-n)!}\right] \tag 2 $$ und für $l\to\infty$, $x = p(l - 1) + t\sqrt{2pq (l-1)}$ die {\it Hermite}schen Polynome $$ \text{const }e^{t^2} \frac{d^n}{dt^n} (e^{-t^2}) \tag 3 $$ hervorgehen. Es sei bemerkt, daß die Polynome (2) vom Standpunkt der zu (1) analogen Orthogonalität eingehend von {\it Charlier, Jordan} und insbesondere {\it H. Pollaczek-Geiringer} [Z. Angew. Math. 8, 292--309 (1928; JFM 54.0559.02), besonders S. 301 u. folg.] untersucht worden sind. Sie spielen in der Statistik eine erhebliche Rolle. (IV 3 D.)
Reviewer: Szegö, Prof. G. (Saint Louis)

33C45Orthogonal polynomials and functions of hypergeometric type
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