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:

$$\begin{aligned} \begin{aligned} J^{(\gamma ),(\lambda )}_{k,\mu }(z)=\sum ^\infty _{n=0}\frac{(\gamma )_{n,k}}{{\varGamma }_k(\lambda n+\mu +1)}\frac{(-\,1)^n}{(n!)^2}\left( \frac{z}{2}\right) ^n \end{aligned}, \end{aligned}$$
(1)

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:

$$\begin{aligned} (\gamma )_{n,k}: = \left\{ \begin{array}{ll} \frac{{\varGamma }_k(\gamma +nk)}{{\varGamma }_k(\gamma )} &{} \quad (k\in \mathbb {R};\gamma \in \mathbb {C}\setminus \{0\}) \\ \gamma (\gamma +k)\ldots (\gamma +(n-1)k) &{} \quad (n\in \mathbb {N};\gamma \in \mathbb {C}), \end{array} \right. \end{aligned}$$
(2)

They gave the relation with the classical Euler’s gamma function(see [2, 8]) as:

$$\begin{aligned} {\varGamma }_k(\gamma )=k^{\frac{\gamma }{k}-1}{\varGamma }\left( \frac{\gamma }{k}\right) , \end{aligned}$$
(3)

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:

$$\begin{aligned} \omega ^{\gamma ,\lambda }_{k,\nu ,b,c}(z)=\sum ^\infty _{n=0}\frac{(-\,1)^nc^n(\gamma )_{n,k}}{{\varGamma }_k(\nu +\lambda n+\frac{b+1}{2})}\frac{\left( \frac{z}{2}\right) ^{\nu +2n}}{(n!)^2} \end{aligned}$$
(4)

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:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \\&\quad =\frac{1}{n!} \xi ^{-\alpha } \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \left( T_{\xi }^{\sigma ,s} \right) ^{n} \left\{ \xi ^{\alpha } \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} , \end{aligned} \end{aligned}$$
(5)

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:

$$\begin{aligned}&\exp \left( tT_{\xi }^{\sigma ,s} \right) \left( \xi ^{\beta } f\left( \xi \right) \right) =\xi ^{\beta } \left( 1-\sigma \xi ^{\sigma } t\right) ^{-\left( \frac{\beta +s}{\sigma } \right) } f\left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } \right) , \end{aligned}$$
(6)
$$\begin{aligned}&\quad \exp \left( tT_{\xi }^{\sigma ,s} \right) \left( \xi ^{\alpha -\sigma n} f\left( \xi \right) \right) =\xi ^{\alpha } \left( 1+\sigma t\right) ^{-1+\left( \frac{\alpha +s}{\sigma } \right) } f\left( \xi \left( 1+\sigma t\right) ^{1/\sigma } \right) , \end{aligned}$$
(7)
$$\begin{aligned}&\quad \left( T_{\xi }^{\sigma ,s} \right) ^{n} \left( \xi uv\right) =\xi \sum _{m=0}^{\infty }\left( \begin{matrix}{n} \\ {m}\end{matrix}\right) \left( T_{\xi }^{\sigma ,s} \right) ^{n-m} \left( v\right) \left( T_{\xi }^{\sigma ,1} \right) ^{m} \left( u\right) , \end{aligned}$$
(8)
$$\begin{aligned}&\quad \begin{aligned}&(1+\xi D)(1+\sigma +\xi D)(1+2\sigma +\xi D)(1+3\sigma +\xi D)\ldots (1+(m-1)\sigma +\xi D)\xi ^{\beta -1} \\&=\sigma ^{m} \left( \frac{\beta }{\sigma } \right) _{m} \xi ^{\beta -1}, \end{aligned} \end{aligned}$$
(9)
$$\begin{aligned} \left( 1-\sigma t\right) ^{\frac{-\alpha }{\sigma } } =\left( 1-\sigma t\right) ^{\frac{-\beta }{\sigma } } \sum _{m=0}^{\infty }\left( \frac{\alpha -\beta }{\sigma } \right) _{m} \frac{\left( \sigma t\right) ^{m}}{m!}. \end{aligned}$$
(10)

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:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) }\left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \xi ^{-\sigma n} t^{n} \\&\quad =\left( 1-\sigma t\right) ^{-\left( \frac{\alpha +s}{\sigma } \right) } \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i}\left( \xi \right) \right] \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \left( 1-\sigma t\right) ^{-1/\sigma } \right) \right] . \end{aligned} \end{aligned}$$
(11)

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:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{n} \\&\quad = \xi ^{-\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \exp (tT_{\xi }^{\sigma ,s} )\left\{ \xi ^{\alpha } \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_1} \left( \xi \right) \right] \right\} , \end{aligned} \end{aligned}$$
(12)

employing the operational technique given in Eq. (6), the above Eq. (12) reduces to:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{n} \\&\quad =(1-\sigma \xi ^{\sigma } t)^{-\left( {\frac{\alpha + s}{ \sigma }} \right) }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi (1-\sigma \xi ^{\sigma } t)^{-1/\sigma } \right) \right] , \end{aligned} \end{aligned}$$
(13)

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:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha -\sigma n \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \xi ^{-\sigma n} t^{n} \\&\quad =\left( 1+\sigma t\right) ^{-1+\left( \frac{\alpha +s}{\sigma } \right) }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \left( 1+\sigma t\right) ^{1/\sigma } \right) \right] . \end{aligned} \end{aligned}$$
(14)

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:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }\xi ^{-\sigma n}V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha -\sigma n \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{n} \\&\quad =\xi ^{-\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \exp \left( tT_{\xi }^{\sigma ,s} \right) \left\{ \xi ^{\alpha -\sigma n}\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} ,\end{aligned} \end{aligned}$$
(15)

applying the operational technique given in Eq. (7), the above Eq. (15) reduces to

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }\xi ^{-\sigma n} V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha -\sigma n \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{n} \\&\quad =\left( 1+\sigma t\right) ^{\frac{\alpha + s}{\sigma } -1}\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \left( 1+\sigma t\right) ^{1/\sigma } \right) \right] , \end{aligned} \end{aligned}$$
(16)

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:

$$\begin{aligned} \begin{aligned}&\sum _{m=0}^{\infty }\left( \begin{matrix} {m+n} \\ {n}\end{matrix}\right) V_{m+n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \xi ^{-\sigma m} t^{m} \\&\quad =\left( 1-\sigma t\right) ^{-\left( {\frac{\alpha +s}{\sigma }} \right) } \frac{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \left( 1-\sigma t\right) ^{-1/\sigma } \right) \right] } \\&\qquad \times \,V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi \left( 1-\sigma t\right) ^{-1/\sigma } ;\sigma ,k_1,\ldots ,k_r,s\right) . \end{aligned} \end{aligned}$$
(17)

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:

$$\begin{aligned} \left( T_{\xi }^{\sigma ,s} \right) ^{n} \left[ \xi ^{\alpha }\displaystyle \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right] =n!\xi ^{\alpha } \frac{V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i}\left( \xi \right) \right] },\nonumber \\ \end{aligned}$$
(18)

multiplying both sides of the above Eq. (18) by \(\exp \left( t\left( T_{\xi }^{\sigma ,s} \right) \right) \), we have

$$\begin{aligned} \begin{aligned}&\exp \left( t\left( T_{\xi }^{\sigma ,s} \right) \right) \left\{ \left( T_{\xi }^{\sigma ,s} \right) ^{n} \left[ \xi ^{\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right] \right\} \\&\quad =n!\exp \left( tT_{\xi }^{\sigma ,s } \right) \left[ \xi ^{\alpha } \frac{V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] } \right] \end{aligned} \end{aligned}$$
(19)
$$\begin{aligned} \begin{aligned}&\sum _{m=0}^{\infty }\frac{t^{m}}{m!} \left( T_{\xi }^{\sigma ,s} \right) ^{m+n} \left\{ \xi ^{\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \\&\quad = n!\exp \left( tT_{\xi }^{\sigma ,s} \right) \left\{ \xi ^{\alpha } \frac{V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] } \right\} , \end{aligned} \end{aligned}$$
(20)

employing the operational technique (6), the above Eq.  (20) can be written as:

$$\begin{aligned}&\sum _{m=0}^{\infty }\frac{t^{m} }{m!} \left( T_{\xi }^{\sigma ,s} \right) ^{m+n} \left[ \xi ^{\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right] \nonumber \\&\quad = n!\xi ^{\alpha } \left( 1-\sigma \xi ^{\sigma } t\right) ^{-\left( \frac{\alpha + s}{\sigma } \right) } \frac{ V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } \right) \right] },\nonumber \\ \end{aligned}$$
(21)

now using Eq. (19) in the above Eq. (21), we have:

$$\begin{aligned} \begin{aligned}&\sum _{m=0}^{\infty }\frac{t^{m} \left( m+n\right) !}{m!n!} \xi ^{\alpha } \frac{V_{m+n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] } \\&\quad =\xi ^{\alpha } \left( 1-\sigma \xi ^{\sigma } t\right) ^{-\left( \alpha +\frac{s}{\sigma } \right) } \frac{V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } \right) \right] }, \end{aligned} \end{aligned}$$
(22)

therefore, we can write the above Eq. (22) as:

$$\begin{aligned}&\sum _{m=0}^{\infty }\left( \begin{matrix}{m+n} \\ {n}\end{matrix}\right) V_{m+n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{m}=\left( 1-\sigma \xi ^{\sigma } t\right) ^{-\left( \alpha +\frac{s}{ \sigma } \right) } \nonumber \\&\quad \times \,\frac{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } \right) \right] },\nonumber \\ \end{aligned}$$
(23)

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:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \\&\quad = \sum _{m=0}^{n }\frac{1}{m!} \left( \sigma \xi ^{\sigma } \right) ^{m} \left( \frac{\alpha }{\sigma } \right) _{m} V_{n-m}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;0 \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) . \end{aligned} \end{aligned}$$
(24)

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:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \\&\quad =\frac{1}{n!} \xi ^{-\alpha } \displaystyle \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \left( T_{\xi }^{\sigma ,s} \right) ^{n} \left\{ \xi \xi ^{\alpha -1}\displaystyle \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} , \end{aligned} \end{aligned}$$
(25)

now applying the operational technique (8), we have:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) =\frac{1}{n!} \xi ^{-\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \xi \sum _{m=0}^{\infty }\left( \begin{matrix}{n} \\ {m}\end{matrix}\right) \\&\quad \times \,\left( T_{\xi }^{\sigma ,s} \right) ^{n-m} \left\{ \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \left( T_{\xi }^{\sigma ,1} \right) ^{m} \left( \xi ^{\alpha -1} \right) \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&=\frac{1}{n!} \xi ^{-\alpha }\displaystyle \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \xi \sum _{m=0}^{\infty }\frac{n!}{m!\left( n-m\right) !} \xi ^{\sigma \left( n-m\right) } \\&\quad \times \, \left[ \left( s+\xi D\right) \left( s+\sigma +\xi D\right) \left( s+2 \sigma +\xi D\right) \ldots \left( s+\left( n-m-1\right) \sigma +\xi D\right) \right] \\&\quad \times \, \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \xi ^{\sigma m} \\&\quad \times \, \left[ \left( 1+\xi D\right) \left( 1+\sigma +\xi D\right) \left( 1+2 \sigma +\xi D\right) \ldots \left( 1+\left( m-1\right) \sigma +\xi D\right) \right] \left( \xi ^{\alpha -1} \right) , \end{aligned} \end{aligned}$$
(26)

using the result given in Eq. (9), the above Eq.  (26) reduces to the following form:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) =\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \sum _{m=0}^{n}\frac{1}{m!\left( n-m\right) !} \xi ^{\sigma n} \\&\quad \times \,\prod _{i=0}^{n-m-1}\left( s+i \sigma +\xi D\right) \left\{ \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \sigma ^{m} \left( \frac{\alpha }{\sigma } \right) _{m}. \end{aligned} \end{aligned}$$
(27)

Putting \(\alpha =0\) and replacing n by \(n-m\) in (26), we get:

$$\begin{aligned} \begin{aligned}&V_{n-m}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;0 \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \\&\quad =\frac{1}{\left( n-m\right) !}\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \\&\quad \left( T_{\xi }^{\sigma ,s} \right) ^{n-m} \left\{ \displaystyle \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \end{aligned} \end{aligned}$$
(28)
$$\begin{aligned} \begin{aligned}&\Rightarrow \frac{1}{\left( n-m\right) !} \left( T_{\xi }^{\sigma ,s} \right) ^{n-m} \left\{ \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \\&\quad =\frac{V_{n-m}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;0 \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] }, \end{aligned} \end{aligned}$$
(29)

the above Eq. (29) gives:

$$\begin{aligned} \begin{aligned}&\frac{1}{\left( n-m\right) !} \prod _{i=0}^{n-m-1}\left( s+i \sigma +\xi D\right) \left\{ \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \\&\quad =\xi ^{\sigma \left( m-n\right) } \frac{V_{n-m}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;0 \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) }{\displaystyle \prod \nolimits _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] }, \end{aligned} \end{aligned}$$
(30)

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:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \\&\quad =\sum _{m=0}^{n }\frac{1}{m!} \left( \sigma \xi ^{\sigma } \right) ^{m} \left( \alpha -\frac{\beta }{\sigma } \right) _{m} V_{n-m}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\beta \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) . \end{aligned} \end{aligned}$$
(31)

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:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{n} \\&\quad =\xi ^{-\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \exp \left( tT_{\xi }^{\left( \sigma ,s\right) } \right) \left\{ \xi ^{\alpha }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} , \end{aligned} \end{aligned}$$
(32)

applying the operational technique given in Eq. (6), the Eq. (32) reduced to:

$$\begin{aligned} \begin{aligned}&\sum _{n=0}^{\infty }V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) t^{n} \\&\quad =\left( 1-\sigma \xi ^{\sigma } t\right) ^{-\left( \alpha +\frac{s}{\sigma } \right) }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i} \\&\quad \left[ -p_{k_i}\left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } \right) \right] , \end{aligned} \end{aligned}$$
(33)

applying the result (10); the Eq. (33) gives:

$$\begin{aligned}&\sum _{n=0}^{\infty }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) t^{n} \nonumber \\&\quad =\left( 1-\sigma \xi ^{\sigma } t\right) ^{-\left( \beta +\frac{s}{\sigma } \right) } \sum _{m=0}^{\infty }\left( \alpha -\frac{\beta }{\sigma } \right) _{m} \frac{\left( \sigma \xi ^{\sigma } t\right) ^{m}}{m!}\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \nonumber \\&\qquad \times \, \omega ^{\gamma ,\lambda }_{\mu ,\nu ,b,c}\left[ -p_{k} \left( \xi \left( 1-\sigma \xi ^{\sigma } t\right) ^{-1/\sigma } \right) \right] \nonumber \\&\quad =\sum _{m=0}^{\infty }\left( \alpha -\frac{\beta }{\sigma } \right) _{m} \frac{\left( \sigma \xi ^{\sigma } t\right) ^{m}}{m!} \xi ^{-\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \exp \left( tT_{\xi }^{\sigma ,s} \right) \nonumber \\&\qquad \times \,\left\{ \xi ^{\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \nonumber \\&\quad =\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\left( \alpha -\frac{\beta }{\sigma } \right) _{m} \frac{\left( \sigma \xi ^{\sigma } \right) ^{m} t^{n+m}}{m!n!} \xi ^{-\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \left( T_{\xi }^{\sigma ,s} \right) ^{n} \nonumber \\&\qquad \times \,\left\{ \xi ^{\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} \nonumber \\&\quad =\sum _{n=0}^{\infty }\sum _{m=0}^{n}\left( \alpha -\frac{\beta }{\sigma } \right) _{m} \frac{\left( \sigma \xi ^{\sigma } \right) ^{m} t^{n}}{ m!\left( n-m\right) !} \xi ^{-\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \left( T_{\xi }^{\sigma ,s} \right) ^{n-m} \nonumber \\&\qquad \times \, \left\{ \xi ^{\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} . \end{aligned}$$
(34)

Now equating the coefficient of \(t^{n} \) , we get:

$$\begin{aligned} \begin{aligned}&V_{n}^{\left( \gamma _i,\lambda _i;\mu _i,\nu _i,b_i,c_i;\alpha \right) } \left( \xi ;\sigma ,k_1,\ldots ,k_r,s\right) \\&\quad = \sum _{m=0}^{n}\left( \alpha -\frac{\beta }{\sigma } \right) _{m} \frac{\left( \sigma \xi ^{\sigma } \right) ^{m}}{ m!\left( n-m\right) !} \xi ^{-\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ p_{k_i} \left( \xi \right) \right] \left( T_{\xi }^{\sigma ,s} \right) ^{n-m} \\&\qquad \times \,\left\{ \xi ^{\beta }\prod _{i=1}^r\omega ^{\gamma _i,\lambda _i}_{\mu _i,\nu _i,b_i,c_i}\left[ -p_{k_i} \left( \xi \right) \right] \right\} , \end{aligned} \end{aligned}$$
(35)

employing the result (5) in Eq. (35), we have the desired formula (31). \(\square \)

Concluding Remarks

  1. 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. 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.