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On the hypergeometric differential equation satisfied by \(_{2}F_{0} (\lambda, \mu; - ; *)\). (English) Zbl 1315.33012

Summary: It is known that the hypergeometric series \(_2F_0(\lambda,\mu;-;z)\), \((\lambda,\mu\in\mathbb{R})\) converges only for \(z=0\), unless it is terminating. The aim here is to discuss about the hypergeometric function \(_2F_0(-m,\mu;-;z)\), \(m\in\mathbb{N}\) (the set of positive integers), which is a polynomial in \(z\) of degree \(m\). We denote the polynomials by \(R_m^\alpha(z)\). It reduces to the well-known Hermite polynomials with suitable choice of the parameter \(\mu\). Hence these polynomials have been named generalised Hermite polynomials. A couple of recurrence relations satisfied by these generalised Hermite polynomials have been derived, from which a generating function of these polynomials have been obtained. This generating function leads to a Rodrigues formula for the generalised Hermite polynomials. It is shown that there is a Hilbert space on which the differential operation generated by the generalised Hermite differential equation is formally self-adjoint. However, to determine orthogonality relation between the generalised Hermite polynomials, the parameters are required to be chosen suitably.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34A05 Explicit solutions, first integrals of ordinary differential equations
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