Abstract
Recently, operational techniques have drawn the attention of several researchers in the study of generating relations and summation formulae. In the present paper, here, we introduce a new sequence of functions involving the product of the generalized k-Bessel function. By using the operational techniques, some generating relations and finite summation formulae of the sequence presented here are also established.
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Introduction and Preliminaries
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are important special functions and these are widely used in physics and engineering such as Electromagnetic waves, Heat conduction, rotational flows, signal processing, Diffusion problems Dynamics of floating bodies, etc. Therefore, these are of interest to engineers and physicists as well as mathematicians. In this paper, we aim to introduce a new sequence of functions involving the product of the generalized k-Bessel function to establish the generating relations and summation formulae by using the operational techniques.
Recently, Romero et al. [8] (see, also [1]) introduced the k-Bessel function of the first kind for \(\lambda ,\gamma ,\nu \in \mathbb {C},k\in \mathbb {R} \) and \( \mathfrak {R}(\lambda )>0, \mathfrak {R}(\nu )>0 \) as follows:
where \((\gamma )_{n,k}\) and \({\varGamma }_k(\gamma )\) are k-Pochhemmer symbol and k-gamma function. These are introduced by Diaz and Pariguan [3] and defined as:
They gave the relation with the classical Euler’s gamma function(see [2, 8]) as:
when \( k=1 \), (2) reduces to the classical Pochhammer symbol and Euler’s gamma function, respectively (see [6]).
In terms of the k-Pochhamer symbol \((\gamma )_{n,k}\) defined by (2), we introduce more generalized form of k-Bessel function \(\omega ^{\gamma ,\lambda }_{k,\nu ,b,c}(z)\) as follows:
where \(\lambda ,\gamma ,\nu ,c,b\in \mathbb {C}\) and \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\nu )>0\).
A new sequence of function \(\left\{ V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \right\} ^{\infty }_{n=0}\) is introduced in this paper as:
where \(\displaystyle T_{\xi }^{\sigma ,s} \equiv \xi ^{\sigma } \left( s+\xi D\right) ,D\equiv \frac{d}{dx} \), \(\sigma \) and s are constants, \(k_1,\ldots ,k_r\) are finite and non-negative integer, \(p_{k_i} \left( \xi \right) \) is a polynomial in \(\xi \) of degree \(k_i\) (where \(i=1,\ldots ,r\)) and \(\omega ^{\gamma ,\lambda }_{\mu ,\nu ,b,c}(\xi ) \) is a generalized k-Bessel function, which is defined in (4). \(T_{\xi }^{\sigma ,s}\) is based on the work of Mittal [4], Patil and Thakare [5], Srivastava and Singh [9].
For our investigation the following operational techniques are required:
Generating Relations
In this section, we stablish here some generating relation involving the product of generalized k-Bessel function by employing the operational techniques.
Theorem 1
Let \(\lambda _i,\gamma _i,\nu _i,b_i,c_i\in \mathbb {C}\); \(\alpha ,\mu _i\in \mathbb {R}\); \(\sigma \) and s are constant; such that \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\nu )>0,\mathfrak {R}(\alpha )+s>0,\sigma >0\), then we have the following formula:
where \(p_{k_i}(\xi )\) is a polynomial in \(\xi \) of degree \(k_i\). \(k_i(i=1,\ldots ,r)\) are finite and non-negative integers.
Proof
To prove the result in Eq. (11), we start from new equation of function given in Eq. (5), from this equation we have:
employing the operational technique given in Eq. (6), the above Eq. (12) reduces to:
after replacing t by \(t\xi ^{-\sigma }\) in Eq. (13), we have the desired result (11). \(\square \)
Theorem 2
Let \(\lambda _i,\gamma _i,\nu _i,b_i,c_i\in \mathbb {C}\); \(\alpha ,\mu _i\in \mathbb {R}\); \(\sigma \) and s are constant; such that \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\nu )>0,\mathfrak {R}(\alpha )+s>0,\sigma >0\), then we have the following formula:
where \(p_{k_i}(\xi )\) is a polynomial in \(\xi \) of degree \(k_i\). \(k_i(i=1,\ldots ,r)\) are finite and non-negative integers.
Proof
Again from Eq. (5), we have:
applying the operational technique given in Eq. (7), the above Eq. (15) reduces to
which is desired. \(\square \)
Theorem 3
Let \(\lambda _i,\gamma _i,\nu _i,b_i,c_i\in \mathbb {C}\); \(\alpha ,\mu _i\in \mathbb {R}\); \(\sigma \) and s are constant; such that \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\nu )>0,\mathfrak {R}(\alpha )+s>0,\sigma >0\), then we have the following formula:
where \(p_{k_i}(\xi )\) is a polynomial in \(\xi \) of degree \(k_i\). \(k_i(i=1,\ldots ,r)\) are finite and non-negative integers.
Proof
To obtained the result (17), we can write Eq. (5) as:
multiplying both sides of the above Eq. (18) by \(\exp \left( t\left( T_{\xi }^{\sigma ,s} \right) \right) \), we have
employing the operational technique (6), the above Eq. (20) can be written as:
now using Eq. (19) in the above Eq. (21), we have:
therefore, we can write the above Eq. (22) as:
replacing t by \(t\xi ^{-\sigma } \) in above Eq. (23), this gives the required result (17). \(\square \)
Finite Summation Formulas
Theorem 4
Let \(\lambda _i,\gamma _i,\nu _i,b_i,c_i\in \mathbb {C}\); \(\alpha ,\mu _i\in \mathbb {R}\); \(\sigma \) and s are constant; such that \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\nu )>0,\mathfrak {R}(\alpha )+s>0,\sigma >0\), then we have the following formula:
where \(p_{k_i}(\xi )\) is a polynomial in \(\xi \) of degree \(k_i\). \(k_i(i=1,\ldots ,r)\) are finite and non-negative integers.
Proof
The Eq. (5) can be written as:
now applying the operational technique (8), we have:
using the result given in Eq. (9), the above Eq. (26) reduces to the following form:
Putting \(\alpha =0\) and replacing n by \(n-m\) in (26), we get:
the above Eq. (29) gives:
from the Eqs. (27) and (30), we have the desired result. \(\square \)
Theorem 5
Let \(\lambda _i,\gamma _i,\nu _i,b_i,c_i\in \mathbb {C}\); \(\alpha ,\mu _i\in \mathbb {R}\); \(\sigma \) and s are constant; such that \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\nu )>0,\mathfrak {R}(\alpha )+s>0,\sigma >0\), then we have the following formula:
where \(p_{k_i}(\xi )\) is a polynomial in \(\xi \) of degree \(k_i\). \(k_i(i=1,\ldots ,r)\) are finite and non-negative integers.
Proof
Begins from Eq. (5), which can be written as:
applying the operational technique given in Eq. (6), the Eq. (32) reduced to:
applying the result (10); the Eq. (33) gives:
Now equating the coefficient of \(t^{n} \) , we get:
employing the result (5) in Eq. (35), we have the desired formula (31). \(\square \)
Concluding Remarks
-
1.
If we choose \(b=c=1\) then generalized k-Bessel function reduced to the following form:
$$\begin{aligned} \omega ^{\gamma ,\lambda }_{k,\mu ,1,1}(z)=\left( \frac{z}{2}\right) ^{\mu }\sum ^\infty _{n=0}\frac{(-1)^n(\gamma )_{n,k}}{{\varGamma }_k(\lambda n+\mu +1)}\frac{\left( \frac{z^2}{4}\right) ^{n}}{(n!)^2}=\left( \frac{z}{2}\right) ^{\mu }J^{(\gamma ),(\lambda )}_{k,\mu }\left( \frac{z^2}{2}\right) , \end{aligned}$$(36)where \(\lambda ,\gamma ,\mu ,\in \mathbb {C}\) and \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\mu )>0\). All the results in Section 2 reduced to involving the product of \(J^{(\gamma ),(\lambda )}_{k,\mu }\left( .\right) \).
-
2.
If we choose \(b=-\,1,c=1\) then generalized k-Bessel function reduced to the k-Wright function [7] associated with the following relation:
$$\begin{aligned} \omega ^{\gamma ,\lambda }_{k,\mu ,-1,1}(z)=\left( \frac{z}{2}\right) ^{\mu }\sum ^\infty _{n=0}\frac{(-1)^n(\gamma )_{n,k}}{{\varGamma }_k(\lambda n+\mu )}\frac{\left( \frac{z^2}{4}\right) ^{n}}{(n!)^2}=\left( \frac{z}{2}\right) ^{\mu }W^{\gamma }_{k,\lambda ,\mu }\left( \frac{-z^2}{2}\right) \end{aligned}$$(37)where \(\lambda ,\gamma ,\mu ,\in \mathbb {C}\) and \(\mathfrak {R}(\lambda )>0,\mathfrak {R}(\mu )>0\). All the results in Section 2 reduced to the involving the product of \(W^{\gamma }_{k,\lambda ,\mu }\left( \frac{-z^2}{2}\right) \).
In this way, with the help of our main sequence formula, some generating relations and finite summation formula of the sequence are also established in the present paper. A new sequence of functions is important due to presence of generalized k-Bessel function \(\omega ^{\gamma ,\lambda }_{k,\nu ,b,c}(z)\). On account of the most general nature of the generalized k-Bessel function \(\omega ^{\gamma ,\lambda }_{k,\nu ,b,c}(z)\) a large number of sequences, generating relations and summation formulae involving simpler functions can be easily obtained as their special cases by assigning the values to the parameters.
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Chand, M., Agarwal, P. & Hammouch, Z. Certain Sequences Involving Product of k-Bessel Function. Int. J. Appl. Comput. Math 4, 101 (2018). https://doi.org/10.1007/s40819-018-0532-8
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DOI: https://doi.org/10.1007/s40819-018-0532-8