GENERALIZED HERMITE POLYNOMIALS
IN THE DESCRIPTION OF
Clemente Cesarano
CHEBYSHEV-LIKE POLYNOMIALS
May 31th, 2023
Vasyl Stefanyk Precarpathian National University�Ivano-Frankivsk, Ukraine�
Clemente Cesarano
Section of Mathematics Luciano Modica
UNINETTUNO University, Roma, Italy
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.
Topics
Introduction
- translation operators
- Weyl rule
- Hausdorff identity
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
produces a shift of the variable
i.e.
further identities can be derived
generalized translation operators
where
are real functions, such that
(invertible)
disentangling rules
the compensation through the commutator
explanatory example
the Hausdorff identity and applications
where
generic operators, independent to
in general,
in the case
i.e.
which gives the generalization
and moreover, for an analytic function
in particular
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
Generalized Hermite Polynomials
- generalized Hermite polynomials
- operational relations
- Monomiality Principle
- multi-index Hermite polynomials
The two-variable Hermite polynomials can be defined by using the relations involving the shift operators
by noting that
we have
by assuming
we obtain
and we can introduce an elementary form of Hermite polynomials
Two-variable Hermite polynomials and ordinary (one-variable) Hermite polynomials
where
partial differential equation
since
the fundamental operational relation
Hermite differential difference equation
Generating function
different classes of generalized Hermite polynomials
by using the following differential difference equation
by manipulating directly the generating function
relevant recurrence relations
give the following partial differential equation
and then
since
finally, a further operational identity
Burchnall-type identities
recurrence relations
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
By setting
the generalized Hermite polynomials are quasi-monomials, and then
A further class of generalized Hermite polynomials
generating function
setting m=2, we get
where the related differential equation is
and, since
the following operational identity follows
they can also recognized as quasi-monomials
and, since
we get
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
[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
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
Two-index, two-variable Hermite polynomials as vectorial extension of the ordinary Hermite
positive quadratic form
a, b, c, real numbers
associated invertible matrix
Hermite polynomials of type
generating function
vectors of space
quadratic form
adjunct quadratic form
two-index, two-variable associated Hermite polynomials of type
where
such that
recurrence relations
where a, b and c are the terms of the quadratic form
recurrence relations
where a, b and c are the terms of the quadratic form
shift operators
with
such that
Partial differential equation
where
orthonormal Hermite functions
since
recurrence relations
can be manipulating to give
shift operators, independent of the discrete parameters
which act
as for the Hermite polynomials, we have
give the following differential equation
we introduce two-variable Hermite functions, by using directly the previous orthonormal one-variable Hermite functions
so that
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
Two-index, two-variable Hermite polynomials and their associated, allow us to define the following functions
that are, in particular, bi-orthonormal
recurrence relations for the bi-orthogonal Hermite functions
the similar structure showed for the one-variable Hermite polynomials and the related orthogonal Hermite functions
further recurrence relations for the bi-orthogonal Hermite functions, by following the similar procedure showed before
so that, we can introduce the relevant shift operators
with
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
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
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
First and second kind Chebyshev polynomials
the Chebyshev complex quantity
so that
Generating functions
since
we obtain
further generating functions
and
Products of Chebyshev polynomials
since
from the Chebyshev complex quantity, we can introduce the related conjugate
further relations
Integral representations by using Hermite polynomials
explicit forms of first and second kind Chebyshev polynomials
by noting that
we have
Integral representations
and similarly
Further Chebyshev polynomials, by using the integral represenations
allows us to state the following recurrence relations
Generalized second kind Chebyshev polynomials
parameter
relevant recurrence relations
Partial differential equation
since
Further generalizations of Chebyshev polynomials, by using the integral representation
Chebyshev polynomials as particular case of Gegenbauer polynomials
Integral representations of Gegenbauer polynomials
which allows to define the following generalization
Recurrence relations involving Chebyshev and Gegenbauer polynomials
Generalized, two-variable second kind Chebyshev polynomials
integral representation
a relevant link between the two families of generalized Chebyshev polynomials
since
we get
Generalized, two-variable first kind Chebyshev polynomials, directly by integral representation
and related recurrence relations
Generating function for the two-variable Chebyshev polynomials
we obtain
since
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
Chebyshev-like Polynomials
- m-order second kind Chebyshev polynomials
- m-order Gegenbauer polynomials
- integral representations
- exotic Hermite polynomials and related Chebyshev polynomials
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
and, similarly the m-order Gegenbauer polynomials
where
particular case for
Generating function
m-order Gegenbauer polynomials
Further generalizations of two-variable Chebyshev polynomials
integral representation
Relevant identities, by using the multi-index Hermite polynomials
we have
Differential recurrence relations
Partial differential recurrence relations
a new class of Hermite polynomials
where
reduce to the ordinary Hermite polynomials by the following operational rule through the confluent hypergeometric function
integral representation
and, since
we get
moreover, by assuming |t|<1
since
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
grazie