In this paper we introduce a generalization of the Voigt functions and discuss their properties and applications. Some interesting explicit series representations, integrals and identities and their link to Jacobi,Laguerre and Hermite polynomials are obtained. The resulting formulas allow a considerable unification of various special results which appear in the literature. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.
En este artículo se introduce una generalización de las funciones de Voigt y se discuten sus propiedades y aplicaciones. Se obtienen representaciones explícitas de series, integrales e identidades y sus conexiones con los polinomios de Jacobi, Laguerre y Hermite. Las fórmulas resultantes permiten la unificación de algunos resultados especiales que aparecen en la literatura. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.
The Voigt functions
where ψ_{2} denotes one of Humbert's confluent hypergeometric function of two variables, defined by
(λ)_{n} being the Pochhammer symbol defined (for λ Ε C) by
The classical Bessel function
so that
Observe that
_{
p
}
The following hypergeometric representation for the Jacobi polynomials
Another special case [
where
The generalized Hermite polynomials (known as Gould-Hopper polynomials)
are 2-variable Kampe de Feriet generalization of the Hermite polynomials Dattoli,
These polynomials usually defined by the generating function
reduce to the ordinary Hermite polynomials
We recall that the Hermite numbers
Where
The special case when
The present work is inspired by the frequent requirements of various properties of Voigt functions in the analysis of certain applied problems. In the present paper it will be shown that generalized Voigt function is expressible in terms of a combination of Kampe de Feriet's functions. We also give further generalizations (involving multivariables) of Voigt functions in terms of series and integrals which are specially useful when the parameters take on special values. The results of multivariable Hermite polynomials are used with a view to obtaining explicit representations of generalized Voigt functions. Our aim is to further introduce two more generalizations of (1.1) and another interesting explicit representation of (1.1) in terms of Kampe de Feriet series
In an attempt to generalize (1.1), we first investigate here the generalized Voigt function
Denition The generalized Voigt function
where
A fairly wide variety of Voigt functions can be represented in terms of the special cases of (2.1).We list below some cases.
The generalized Voigt function
is defined by the integral representation
where
An obvious special case of (2.1) occurs when we take
Clearly, the case
and (2.2) corresponds to (1.1) and (1.2) and we have
And
Moreover,
Using the denition (2.1) with
we get a connection between
where V_{μ},_{V} is given by (1.1).
Similarly setting
we get
In (2.1),we expand _{
1
}
which may be rewritten in the form
where we have used the series manipulation [
By using a well-known Kummer's theorem [
in (2.1) yields
which further for
For
In view of the result (1.7)[
We consider the formula [
which on replacing
And
respectively. These last two results are now applied to (2.1) to yield a double series representation
As before, set r = 2 and use
Putting X=0 and using the property
Now consider a result [
which on replacing
And
respectively. These last two results are now applied to (2.1) to yield an integral representation
The use of generalized Hermite polynomials defined by (1.8) can be exploited to obtain the series representations of (2.1). We have indeed
by applying (1.8) to the integral on the right of (2.1). Since
we may write a limiting case of (2.1) in the form
which further for X=0 reduces to
Now in (4.6), using [
where
A reduction of interest involves the case of replacing
We consider the following two integrals
where H_{v} (x) are Struve functions [
where s_{λ,v} (x) are Lommel functions [
To evaluate these two integrals,we will apply the following two results [
Making appropriate substitution of H_{v} (x) and S_{λν} (x) from these two results in (5.1) and (5.2), we get
For X=0, (5.1) and (5.2) reduce to
Setting r=2 and
Setting r=2 in (5.8) and using [
where α =
First we consider a number which we denote by
The series expansion for
On comparing (6.1) with (1.14),we find that the number
Moreover from (6.1),we can obtain the following two Laplace transforms
where Ψ is logarithmic derivative of Γ function [
Now we start with a result [
which on replacing
On multiplying these two results yields
which is equivalent to
Using (6.1) and (1.10) in (6.4) gives
Comparing the coecients of
In view of the result (1.12) expressed for Hermite numbers H_{n}, for
Now we turn to the derivation of the representation of voigt function from (6.7). Multiply both he sides of (6.4) by
which on using (6.1) gives
For
Yet, another immediate consequence of (6.9) is obtained by taking y1 =
By setting z=0 in (6.4) and multiplying both he sides by
we get
where Ψ is logarithmic derivative of Γ function [
If, in (6.5),we set
Now we start with a result [
which on replacing
On multiplying these two results and adjusting the variables yields
which is equivalent to
Using the definition of Lagrange-Hermite polynomials
which on replacing n by n-2m gives
Again applying the denition of Hermite polynomials given by (1.10) in (7.4), replacing n by n-k and comparing the coecients of
which reduces to (6.7) when we take x1 = x2 = 1 and use
It is also fairly straightforward to get a representation of generalized Voigt function
On the other hand, multiplying both the sides of (7.4) by
and integrating with respect to t from 0 to ∞and then using (2.2), we get a generalization of (6.10) in the form