A limit relationship between Laguerre and Hermite polynomials

In this paper, for the proposed extended Laguerre polynomials A α q , n x , the generalized hypergeometric function of the type F q q , q > 2 and extension of the Laguerre polynomial are introduced. Similar to those related to the Laguerre polynomials, the generating function, recurrence relations, and Rodrigue’s formula are determined. Some corollaries are also discussed at the end.

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“…Chen and Srivastava [5] found a stronger Rodrigues formula to develop a generalization of the Laguerre polynomial.…”

Section: Introduction and Applicationsmentioning

confidence: 99%

The Extended Laguerre Polynomials $\{A_{q, n}\}$

Khan

Kalim

Akgül

et al. 2022

Journal of Function Spaces

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“…For example, the Hermite polynomials described by 2 F 0 hypergeometric functions are located at the lowest level of the scheme, and the generalized Laguerre polynomials located on next level of the scheme are connected to the Hermite polynomials both via an exact limit relation [2] (9.12.13) and through a well-known special case that is separately valid for even and odd polynomials [2] (p. 244) (see Equations ( 16) and (19) in this paper). Some other interesting properties of these polynomials are a general limit relation between these two polynomials in terms of the Srivastava-Singhal polynomials [3] and the appearance as a generic polynomial solution of a differential equation [4]. In general, one easily observes from the Askey table that all orthogonal polynomials within this scheme are special or limiting cases of the Wilson polynomials or Racah polynomials (or Askey-Wilson or q-Racah polynomials).…”

Section: Introductionmentioning

confidence: 99%

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

2021

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In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.

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“…There have been many studies on polynomials via the Rodrigues formula [3][4][5][6][7] and new papers constantly coming out related with these generalizations [8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Some properties of generalized multiple Hermite polynomials

Kaanoğlu¹,

Özarslan²

2011

Journal of Computational and Applied Mathematics

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a b s t r a c tThe purpose of this paper is to introduce and discuss a more general class of multiple Hermite polynomials. In this work, the explicit forms, operational formulas and a recurrence relation are obtained. Furthermore, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple Hermite polynomials.

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A limit relationship between Laguerre and Hermite polynomials

Cited by 8 publications

References 3 publications

The Extended Laguerre Polynomials $\{A_{q, n}\}$

The Extended Laguerre Polynomials $\{A_{q, n}\}$

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

Some properties of generalized multiple Hermite polynomials

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A limit relationship between Laguerre and Hermite polynomials

Cited by 8 publications

References 3 publications

The Extended Laguerre Polynomials A q , n

The Extended Laguerre Polynomials A q , n

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

Some properties of generalized multiple Hermite polynomials

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Product

Resources

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The Extended Laguerre Polynomials $\{A_{q, n}\}$

The Extended Laguerre Polynomials $\{A_{q, n}\}$