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A generalization of the Hermite polynomials. (Spanish. English summary) Zbl 0770.33006

The authors define the polynomials \(H_{2n+r,\alpha}(x)\), \(n=0,1,\dots,\alpha>-1\) and \(r\) a fixed non-negative integer, by the relation \[ H_{2n+r,\alpha}(x)={(-1)^ n(2n+r)!\over n!} (2x)^ r\Phi(-n,\alpha+1;x^ 2), \] where \(\Phi\) is the confluent hypergeometric function. If \(r=0\) and \(\alpha=-1/2\) and \(\alpha=1/2\) these polynomials reduce to the classical Hermite polynomials \(H_{2n}(x)\) or \(H_{2n+1}(x)\) respectively. Series expansion, generating function, recurrence relation, orthogonality property and other properties of the polynomials \(H_{2n+r,\alpha}(x)\) are obtained.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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