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GENERALIZED HERMITE POLYNOMIALS

IN THE DESCRIPTION OF

Clemente Cesarano

CHEBYSHEV-LIKE POLYNOMIALS

May 31th, 2023

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Vasyl Stefanyk Precarpathian National University�Ivano-Frankivsk, Ukraine

Clemente Cesarano

Section of Mathematics Luciano Modica

UNINETTUNO University, Roma, Italy

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This presentation is a survey on the description of the main properties of the multi-index or multi-dimensional Chebyshev polynomials, by using the generalized Hermite polynomials as tool. The Hermite polynomials play a fundamental role in the extension of the classical special functions to the multidimensional or multi-index case. We will also show that, starting from the multi-index Hermite polynomials, it is possible to introduce the Chebyshev polynomials of multidimensional type of first and second kind, and some of their generalizations.

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  • Introduction

Topics

  • Generalized Hermite polynomials
  • Bi-orthogonal Hermite functions
  • Chebyshev polynomials and Integral Representations
  • Chebyshev-like polynomials

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Introduction

- translation operators

- Weyl rule

- Hausdorff identity

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Translation Operators

a real function which is analytic in a neighborhood of the origin, can be expanded in Taylor series

the so called shift or translation operator

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produces a shift of the variable

i.e.

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further identities can be derived

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generalized translation operators

where

are real functions, such that

(invertible)

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disentangling rules

the compensation through the commutator

explanatory example

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the Hausdorff identity and applications

where

generic operators, independent to

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in general,

in the case

i.e.

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which gives the generalization

and moreover, for an analytic function

in particular

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References

[1] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge Univerity Press, London, 1958

[2] H.M. Srivastava and H.L. Manocha, A treatise on Generating Functions, Ellis Horwood, New York, 1984

[3] Y. Smirnov and A. Turbiner, Hidden SL-2 Algebra of finite-difference equations, Modern Phys. Lett A10 (1995), 1795-1801

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Generalized Hermite Polynomials

- generalized Hermite polynomials

- operational relations

- Monomiality Principle

- multi-index Hermite polynomials

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The two-variable Hermite polynomials can be defined by using the relations involving the shift operators

by noting that

we have

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by assuming

we obtain

and we can introduce an elementary form of Hermite polynomials

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Two-variable Hermite polynomials and ordinary (one-variable) Hermite polynomials

where

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partial differential equation

since

the fundamental operational relation

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Hermite differential difference equation

Generating function

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different classes of generalized Hermite polynomials

by using the following differential difference equation

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by manipulating directly the generating function

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relevant recurrence relations

give the following partial differential equation

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and then

since

finally, a further operational identity

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Burchnall-type identities

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recurrence relations

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Monomiality Principle and Hermite polynomials

a given polynomial

can be considered quasi-monomial, if two operators can be defined in a such way that

with

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By setting

the generalized Hermite polynomials are quasi-monomials, and then

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A further class of generalized Hermite polynomials

generating function

setting m=2, we get

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where the related differential equation is

and, since

the following operational identity follows

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they can also recognized as quasi-monomials

and, since

we get

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References

[4] P. Appell, J.Kampé De Fériet, Fonctions hypergéométriques et hypersphériques: Polynomes d’Hermite; Gauthier-Villars, Paris, France, 1926

[5] H.W. Gould, A. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. (1962), vol. 29, pp. 51–63

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[6] G. Dattoli, P.E. Ricci, C. Cesarano, The Bessel Functions and the Hermite Polynomials from a unified point of view, APPLICABLE ANALYSIS, (2001) vol. 80, pp. 379-384

[7] G. Dattoli, S. Lorenzutta, C. Cesarano, Generalized Polynomials and New Families of Generating Functions, Annali dell’Università di Ferrara, Sez. 7 Scienze Matematiche, (2001) vol. XLVII, pp. 57-61

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Bi-orthogonal Hermite Functions

- two-index, two-variable Hermite polynomials

- relevant recurrence relations

- orthogonal Hermite functions of one and two variables

- bi-orthogonal Hermite functions

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Two-index, two-variable Hermite polynomials as vectorial extension of the ordinary Hermite

positive quadratic form

a, b, c, real numbers

associated invertible matrix

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Hermite polynomials of type

generating function

vectors of space

quadratic form

adjunct quadratic form

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two-index, two-variable associated Hermite polynomials of type

where

such that

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recurrence relations

where a, b and c are the terms of the quadratic form

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recurrence relations

where a, b and c are the terms of the quadratic form

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shift operators

with

such that

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Partial differential equation

where

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orthonormal Hermite functions

since

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recurrence relations

can be manipulating to give

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shift operators, independent of the discrete parameters

which act

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as for the Hermite polynomials, we have

give the following differential equation

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we introduce two-variable Hermite functions, by using directly the previous orthonormal one-variable Hermite functions

so that

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Condition of bi-orthogonality for the vectorial Hermite polynomials

suggest to introduce similar Hermite functions; not orthogonal functions, but satisfied the property of bi-orthogonality

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Two-index, two-variable Hermite polynomials and their associated, allow us to define the following functions

that are, in particular, bi-orthonormal

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recurrence relations for the bi-orthogonal Hermite functions

the similar structure showed for the one-variable Hermite polynomials and the related orthogonal Hermite functions

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further recurrence relations for the bi-orthogonal Hermite functions, by following the similar procedure showed before

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so that, we can introduce the relevant shift operators

with

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Finally, we obtain the following partial differential equation solved by the bi-orthogonal Hermite functions

Where, similarly as for the two-index, two-variable Hermite polynomials

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References

[8] M. Bertola, M.Gekhtman, J. Szmigielski, Cauchy biorthogonal polynomials, J. Approx. Theory (2010), vol. 162, pp. 832–867

[9] C. Cesarano, A Note on Bi-Orthogonal Polynomials and Functions, FLUIDS (2020), vol. 5, pp. 1-15

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Chebyshev Polynomials�and Integral Representations

- ordinary Chebyshev polynomials in terms of complex quantity

- products of Chebyshev polynomials

- integral representations ofChebyshev polynomials

- generalized two-variable Chebyshev polynomials

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First and second kind Chebyshev polynomials

the Chebyshev complex quantity

so that

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Generating functions

since

we obtain

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further generating functions

and

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Products of Chebyshev polynomials

since

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from the Chebyshev complex quantity, we can introduce the related conjugate

further relations

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Integral representations by using Hermite polynomials

explicit forms of first and second kind Chebyshev polynomials

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by noting that

we have

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Integral representations

and similarly

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Further Chebyshev polynomials, by using the integral represenations

allows us to state the following recurrence relations

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Generalized second kind Chebyshev polynomials

parameter

relevant recurrence relations

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Partial differential equation

since

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Further generalizations of Chebyshev polynomials, by using the integral representation

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Chebyshev polynomials as particular case of Gegenbauer polynomials

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Integral representations of Gegenbauer polynomials

which allows to define the following generalization

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Recurrence relations involving Chebyshev and Gegenbauer polynomials

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Generalized, two-variable second kind Chebyshev polynomials

integral representation

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a relevant link between the two families of generalized Chebyshev polynomials

since

we get

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Generalized, two-variable first kind Chebyshev polynomials, directly by integral representation

and related recurrence relations

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Generating function for the two-variable Chebyshev polynomials

we obtain

since

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References

[10] C. Cesarano, Identities and generating functions on Chebyshev polynomials, Georgian Math. J. (2012), vol. 19, pp.427–440

[11] C. Cesarano, Generalized Chebyshev polynomials, Hacet. J. Math. Stat. (2014), vol. 43, pp. 731–740.

[12] C. Cesarano, Integral representations and new generating functions of Chebyshev polynomials, Hacet. J. Math. Stat (2015), vol. 44, pp. 535–546

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Chebyshev-like Polynomials

- m-order second kind Chebyshev polynomials

- m-order Gegenbauer polynomials

- integral representations

- exotic Hermite polynomials and related Chebyshev polynomials

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By using the generalized m-order Hermite polynomials, we can obtain further generalizations of Chebyshev polynomials which present a substantial generalization of the ordinary Chebyshev polynomials, then could be called as Chebyshev-like polynomials

m-order second kind Chebyshev polynomials

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and, similarly the m-order Gegenbauer polynomials

where

particular case for

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Generating function

m-order Gegenbauer polynomials

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Further generalizations of two-variable Chebyshev polynomials

integral representation

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Relevant identities, by using the multi-index Hermite polynomials

we have

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Differential recurrence relations

Partial differential recurrence relations

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a new class of Hermite polynomials

where

reduce to the ordinary Hermite polynomials by the following operational rule through the confluent hypergeometric function

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integral representation

and, since

we get

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moreover, by assuming |t|<1

since

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References

[13] M. Abramowitz and I. A. Stegun, Eds., Dover Books on Advanced Mathematics, Dover Publications, New York, 1965

[14] A. Wünsche, Generalized Hermite polynomials associated with functions of parabolic cylinder, Appl. Math. Comput. (2003), vol. 141(1), pp. 197-213

[15] G. Dattoli, C. Cesarano, On a new family of Hermite polynomials associated to parabolic cylinder functions, Appl. Math. Comput. (2003), vol. 141(1), pp. 143-149

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grazie