An Integral Relation Associated with a General Class of Polynomials and the Aleph Function

Abstract

A new finite integral involving two general class of polynomials with the Aleph function has been obtained in the present paper. This integral is supposed to be one of the most universal integral evaluated until now and act as a key component from which we can obtain as its different special cases, integrals relating a large number of simpler special functions and polynomials. Some useful unique cases of the main outcome have also been discussed in the paper.

1 Introduction

The Aleph-function is a new generalization of the well-known H-function [1] and the I-function [2, 3].

The Aleph-function is defined and represented as follows [4, 5].

$$ \begin{aligned} \aleph [{\text{z}}] & = \aleph_{{{\text{P}}_{\text{i}} ,{\text{Q}}_{\text{i}} ,\tau_{\text{i}} ;{\text{r}}}}^{\text{M,N}} [{\text{z}}] = \aleph_{{{\text{P}}_{\text{i}} ,{\text{Q}}_{\text{i}} ,\tau_{\text{i}} ;{\text{r}}}}^{\text{M,N}} \left[ {{\text{z }}\left| {_{{ ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\,[\tau_{\text{i}} ({\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )]_{{{\text{M}} + 1,{\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )_{{ 1 , {\text{N}}}} ,\, \ldots ,\,[\tau_{\text{i}} ({\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )]_{{{\text{N}} + 1,{\text{P}}_{\text{i}} }} }} } \right.} \right] \\ & = \frac{1}{{2\uppi \upomega }}\int\limits_{\text{L}} {\Phi (\xi ){\text{z}}^{ - \xi } } {\text{d}}\xi \\ \end{aligned} $$

(1.1)

for all z ≠ 0, where $\upomega = \sqrt { - 1}$ and

$$ \Phi (\xi ) = \frac{{\prod\nolimits_{{{\text{j}} = 1}}^{\text{M}} {\Gamma ({\text{b}}_{\text{j}} + {\text{B}}_{\text{j}} \xi )} \prod\nolimits_{{{\text{j}} = 1}}^{\text{N}} {\Gamma (1 - {\text{a}}_{\text{j}} - {\text{A}}_{\text{j}} \xi )} }}{{\sum\nolimits_{{{\text{i}} = 1}}^{\text{r}} {\tau_{\text{i}} } \prod\nolimits_{{{\text{j}} = {\text{N}} + 1}}^{{{\text{P}}_{\text{i}} }} {\Gamma ({\text{a}}_{\text{ji}} + {\text{A}}_{\text{ji}} \xi )} \prod\nolimits_{{{\text{j}} = {\text{M}} + 1}}^{{{\text{Q}}_{\text{i}} }} {\Gamma (1 - {\text{b}}_{\text{ji}} - {\text{B}}_{\text{ji}} \xi )} }} $$

(1.2)

The path of integration ${\text{L}} = {\text{L}}_{{{\text{i}}\Upsilon \infty }} ,\Upsilon \in {\text{R}}$ extends from $\varUpsilon$  − i∞ to $\varUpsilon$  + i∞. The poles of $\Gamma ({\text{b}}_{\text{j}} + {\text{B}}_{\text{j}} \xi ),{\text{ j}} = \, \overline{{ 1 , {\text{M}}}} .$ which do not coincide to the poles of $\Gamma (1 - {\text{a}}_{\text{j}} - {\text{A}}_{\text{j}} \xi ),{\text{j}} = \overline{{ 1 , {\text{N}}}}$ are taken as simple poles. The parameters p_i, q_i are non-negative integers 0 ≤ N ≤ P_i, 1 ≤ M ≤ Q_i, τ_i > 0 for ${\text{i}} = \overline{{ 1 , {\text{r}}}}$. The parameters ${\text{A}}_{\text{j}} ,{\text{B}}_{\text{j}} ,{\text{A}}_{\text{ji}} ,{\text{B}}_{\text{ji}} > 0\;{\text{and}}\;{\text{a}}_{\text{j}} ,{\text{b}}_{\text{j}} ,{\text{a}}_{\text{ji}} ,{\text{b}}_{\text{ji}} \in {\text{C}}$. The product in (1.2) is interpreted as unity. The existence conditions for the described integral (1.1) are given beneath:

$$ \uptheta_{\ell } > 0,\;\left| {\arg ({\text{z}})} \right| < \frac{\pi }{2}\uptheta_{\ell } ,\ell = \overline{{1,{\text{r}}}} ; $$

(1.3)

$$ \uptheta_{\ell } > 0,\left| {\arg ({\text{z}})} \right| < \frac{\pi }{ 2}\uptheta_{\ell } \;{\text{and}}\;\text{Re} \{ \zeta_{\ell } \} + 1 < 0, $$

(1.4)

where

$$ \theta_{\ell } = \sum\limits_{{{\text{j}} = 1}}^{\text{N}} {{\text{A}}_{\text{j}} } + \sum\limits_{{{\text{j}} = 1}}^{\text{M}} {{\text{B}}_{\text{j}} - \tau_{\ell } } \left( {\sum\limits_{{{\text{j}} = {\text{N}} + 1}}^{{{\text{P}}_{\ell } }} {{\text{A}}_{{{\text{j}}\ell }} } + \sum\limits_{{{\text{j}} = {\text{M}} + 1}}^{{{\text{Q}}_{\ell } }} {{\text{B}}_{{{\text{j}}\ell }} } } \right) $$

(1.5)

$$ \zeta_{\ell } = \sum\limits_{{{\text{j}} = 1}}^{\text{M}} {{\text{b}}_{\text{j}} } - \sum\limits_{{{\text{j}} = 1}}^{\text{N}} {{\text{a}}_{\text{j}} + \tau_{\ell } } \left( {\sum\limits_{{{\text{j}} = {\text{M}} + 1}}^{{{\text{Q}}_{\ell } }} {{\text{b}}_{{{\text{j}}\ell }} } - \sum\limits_{{{\text{j}} = {\text{N}} + 1}}^{{{\text{P}}_{\ell } }} {{\text{a}}_{{{\text{j}}\ell }} } } \right) + \frac{1}{2}({\text{P}}_{\ell } - {\text{Q}}_{\ell } ),\ell = \overline{{1,{\text{r}}}} , $$

(1.6)

Note 1

The simplification of the sum in the denominator of (1.2) in terms of a polynomial in ξ, the factor of this polynomial can be uttered by a fraction of Euler’s Gamma function leading to H-function, see [6], p. 325.

Note 2

It might be seen that there is no recorded name given to (1.1), compared to [5]. The Mellin transform of this function is coefficient of ${\text{z}}^{ - \zeta }$ in the integrand of (1.1).

Note 3

Taking $\uptau_{\text{i}} = 1$, i = 1, …, r, in (1.1), the $\aleph$-function lessens to the notable I-function [3].

Note 4

Putting r = 1 and $\uptau_{1} =\uptau_{2} = \ldots =\uptau_{3} = 1,$ then $\aleph$-function reduces to the known H-function [7].

Following definition of general class of polynomials is required which was introduced by Srivastava [8, Eq. (1)].

$$ {\text{S}}_{\text{n}}^{\text{m}} [{\text{x}}] = \sum\limits_{{{\text{k}} = 0}}^{{ [ {\text{n}}/{\text{m}}]}} {\frac{{( - {\text{n}})_{\text{mk}} }}{{{\text{k}}!}}} {\text{A}}_{{{\text{n}},{\text{k}}}} {\text{x}}^{\text{k}} ,\quad \quad {\text{n}} = 0,1,2, \ldots $$

(1.7)

Here the coefficients ${\text{A}}_{{{\text{n}},{\text{k}}}} ({\text{n}},{\text{k}} \ge 0)$ are subjective real or complex constants, whereas M₁ is an arbitrarily chosen positive integer.

On suitably specializing the coefficients ${\text{A}}_{{{\text{n}},{\text{k}}}}$ occurring in (1.7), the general class polynomials ${\text{S}}_{\text{n}}^{\text{m}} [{\text{x}}]$ can be reduced to the known traditional orthogonal polynomials and the generalized hypergeometric polynomials as its particular cases. These incorporate, among others, the Hermite polynomials, the Jacobi polynomials, the Laguerre polynomials, the Bessel polynomials, the Gould-Hopper polynomials and a couple of others.

2 Main Result

$$ \begin{aligned} & \int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } \\ & \quad .\aleph_{{{\text{P}}_{\text{i}} , {\text{Q}}_{\text{i}} ,\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N}} \left[ {{\text{z}}\left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} )_{{1 , {\text{M}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} , {\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\;{\text{A}}_{\text{j}} )_{{1 , {\text{N}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{a}}_{\text{j}} , {\text{A}}_{\text{j}} ) ]_{{{\text{N}} + 1 , {\text{P}}_{\text{i}} }} }} } \right.} \right] \\ & \quad .{\text{S}}_{{{\text{n}}_{1} }}^{{{\text{m}}_{1} }} \left[ {{\text{z}}_{1} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\upmu} } \right]{\text{S}}_{{{\text{n}}_{2} }}^{{{\text{m}}_{2} }} \left[ {{\text{z}}_{2} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } \right]{\text{dx}} \\ & \quad = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} /{\text{m}}_{1} } \right]}} {\sum\limits_{{{\text{k}}_{ 2} = 0}}^{{\left[ {{\text{n}}_{2} /{\text{m}}_{2} } \right]}} {\frac{{\left( { - {\text{n}}_{1} } \right)_{{{\text{m}}_{1} {\text{k}}_{ 1} }} \left( { - {\text{n}}_{2} } \right)_{{{\text{m}}_{2} {\text{k}}_{ 2} }} }}{{{\text{k}}_{1} !{\text{k}}_{2} !}}} } {\text{A}}_{{{\text{n}}_{1} , {\text{k}}_{ 1} }} \;{\text{A}}_{{{\text{n}}_{2} ,{\text{k}}_{ 2} }} {\text{z}}_{ 1}^{{{\text{k}}_{ 1} }} {\text{z}}_{ 2}^{{{\text{k}}_{ 2} }} \\ & \quad .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} + \left( {\uplambda^{{\prime }} +\upmu^{{\prime }} } \right){\text{k}}_{ 2} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} }} \\ & \quad .\aleph_{{{\text{P}}_{\text{i}} { + }2 , {\text{Q}}_{\text{i}} { + }1,\uptau_{\text{i}} ; {\text{r}}}}^{{{\text{M,N + }}2}} \left[ {{\text{z}}\left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{b}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{a}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\,[\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{\left( {1\, - \,{\text{u}}\, - \,\uplambda{\text{k}}_{1} \, - \,\uplambda^{{\prime }} {\text{k}}_{2} ,s} \right),\left( {1\, - \,{\text{v}}\, - \,\upmu{\text{k}}_{ 1} \, - \,\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{t}}} \right)}} } \right.} \right. \\ \end{aligned} $$

$$ \left. {\begin{array}{*{20}c} {\left( {{\text{a}}_{\text{j}} ,\;\upalpha_{\text{j}} } \right)_{{ 1 , {\text{N}}}} ,\;\tau_{j} \left( {{\text{a}}_{\text{j}} ,\;\upalpha_{\text{j}} } \right)_{{{\text{N}} + 1 , {\text{P}}_{i} ;r}} } \\ {\left( {1 - {\text{u}} - {\text{v}} -\uplambda{\text{k}}_{ 1} -\upmu{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} -\upmu^{{\prime }} {\text{k}}_{ 2} ,{\text{s}} + {\text{t}}} \right)} \\ \end{array} } \right], $$

(2.1)

where s > 0, t > 0, Re (u + s b_j/β_j) > 0, Re (v + t b_j/β_j) > 0,

j = 1, …, M, λ, λ′, μ and μ′ are positive integers. ${\text{A}}_{{{\text{n}}_{1} ,{\text{k}}_{ 1} }} \;{\text{and}}\;{\text{A}}_{{{\text{n}}_{2} ,{\text{k}}_{ 2} }}$ (${\text{n}}_{1} ,\;{\text{k}}_{ 1}$, ${\text{n}}_{2} ,\;{\text{k}}_{ 2}$  ≥ 0) are arbitrary constants, real or complex.

Proof

To establish (2.1), expressing the $\aleph$-function by (1.2) and general class of polynomials by (1.7), then the order of summations and integration are interchanged (which is justified due to the absolute convergence of the integral in the process), we calculate the integral with the help of a result ([7], p. 287 (3.119)), and get the desired outcome.

3 Special Cases

• (A) Taking ${\text{S}}_{\text{n}}^{ 2} \left[ {\text{y}} \right] = {\text{y}}^{{{\text{n}}/2}} {\text{H}}_{\text{n}} \left[ {\frac{1}{{2\sqrt {\text{y}} }}} \right]$ in the result obtained in (2.1) to the case of Hermite polynomials ([9], Eq. (5.5.4), p. 106 and [3], p. 158)

in which case m₁ = 2, ${\text{A}}_{{{\text{n}}_{1} ,{\text{k}}_{ 1} }} = \left( { - 1} \right)^{{{\text{k}}_{ 1} }}$ and also letting m₂ = 2, ${\text{A}}_{{{\text{n}}_{2} ,{\text{k}}_{ 2} }} = \left( { - 1} \right)^{{{\text{k}}_{ 2} }}$, we have

$$ \begin{array}{*{20}l} {\int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } } \hfill \\ { .\aleph_{{{\text{P}}_{\text{i}} , {\text{Q}}_{\text{i}} ,\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N}} \left[ {{\text{z}}\left( {\frac{{{\text{x}}\, - \,{\text{a}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}}\, - \,{\text{x}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\,[\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\;{\text{A}}_{\text{j}} )_{{ 1 , {\text{N}}}} ,\, \ldots ,\,[\uptau_{\text{i}} ( {\text{a}}_{\text{j}} ,\;{\text{A}}_{\text{j}} ) ]_{{{\text{N}} + 1 , {\text{P}}_{\text{i}} }} }} } \right.} \right]} \hfill \\ \end{array} $$

$$ .\left[ {{\text{z}}_{ 1} \left( {\frac{{{\text{x}}\, - \,{\text{a}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}}\, - \,{\text{x}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{\upmu} } \right]^{{{\text{n}}_{1} /2}} {\text{H}}_{{{\text{n}}_{1} }} \left[ {\frac{1}{{2\sqrt {{\text{z}}_{ 1} \left( {\frac{{{\text{x}}\, - \,{\text{a}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}}\, - \,{\text{x}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{\upmu} } }}} \right] $$

$$ .\left[ {{\text{z}}_{ 2} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } \right]^{{{\text{n}}_{2} /2}} {\text{H}}_{{{\text{n}}_{2} }} \left[ {\frac{1}{{2\sqrt {{\text{z}}_{ 2} \left( {\frac{{{\text{x}}\, - \,{\text{a}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}}\, - \,{\text{x}}}}{{{\text{x}}\, - \,{\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } }}} \right]{\text{dx}} $$

$$ = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} /{\text{m}}_{1} } \right]}} {\sum\limits_{{{\text{k}}_{ 2} = 0}}^{{{\text{n}}_{2} /{\text{m}}_{2} }} {\frac{{\left( { - {\text{n}}_{1} } \right)_{{ 2 {\text{k}}_{ 1} }} ( - {\text{n}}_{2} )_{{ 2 {\text{k}}_{ 2} }} }}{{{\text{k}}_{1} !{\text{k}}_{2} !}}} } \left( { - 1} \right)^{{{\text{k}}_{ 1} }} \left( { - 1} \right)^{{{\text{k}}_{ 2} }} {\text{z}}_{ 1}^{{{\text{k}}_{ 1} }} {\text{z}}_{ 2}^{{{\text{k}}_{ 2} }} $$

$$ .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} + \left( {\uplambda^{{\prime }} +\upmu^{{\prime }} } \right){\text{k}}_{ 2} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} }} $$

$$ .\aleph_{{{\text{P}}_{\text{i}} {\text{ + 2,Q}}_{\text{i}} + 1 ,\uptau_{\text{i}} ; {\text{r}}}}^{{{\text{M,}}\,{\text{N + 2}}}} \left[ {{\text{z}}\left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{b}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{a}} - {\text{c}}}}} \right)^{\text{s}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{\left( { 1\, - \,{\text{u}}\, - \,\uplambda{\text{k}}_{ 1} \, - \,\uplambda^{{\prime }} {\text{k}}_{2} ,\,s} \right),\left( {1\, - \,{\text{v}}\, - \,\upmu{\text{k}}_{ 1} \, - \,\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{t}}} \right)}} } \right.} \right. $$

$$ \left. {\begin{array}{*{20}c} {\left( {{\text{a}}_{\text{j}} ,\;\upalpha_{\text{j}} } \right)_{{1,{\text{N}}}} ,\tau_{j} \left( {{\text{a}}_{\text{j}} ,\;\upalpha_{\text{j}} } \right)_{{{\text{N}} + 1 , {\text{P}}_{i} ;r}} } \\ {\left( {1 - {\text{u}} - {\text{v}} -\uplambda{\text{k}}_{ 1} -\upmu{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} -\upmu^{{\prime }} {\text{k}}_{ 2} ,{\text{s}} + {\text{t}}} \right)} \\ \end{array} } \right], $$

(3.1)

applicable under the conditions as available from (2.1).

• (B) For the Jacobi polynomials ([9], Eq. (4.3.2), p. 68 and [3], p. 158), our result (2.1) yields the following result by setting

$$ {\text{S}}_{\text{n}}^{ 1} \left[ {\text{x}} \right] = {\text{P}}_{\text{n}}^{{\left( {\upalpha^{{\prime }} ,\upbeta^{{\prime }} } \right)}} \left( { 1- 2 {\text{x}}} \right)\;{\text{in}}\;{\text{which}}\;{\text{case}} $$

$$ {\text{m}}_{1} = 1 ,\quad {\text{A}}_{{{\text{n}}_{1} , {\text{k}}_{ 1} }} = \left( {\begin{array}{*{20}c} {{\text{n}}_{1} + {\text{k}}_{ 1} } \\ {{\text{n}}_{1} } \\ \end{array} } \right)\frac{{\left( {\upalpha^{{\prime }} +\upbeta^{{\prime }} + {\text{n}}_{1} + 1} \right)_{{{\text{k}}_{ 1} }} }}{{\left( {\upalpha^{{\prime }} + 1} \right)_{{{\text{k}}_{ 1} }} }} $$

and also taking

$$ {\text{m}}_{2} = 1 ,\quad {\text{A}}_{{{\text{n}}_{2} , {\text{k}}_{ 2} }} = \left( {\begin{array}{*{20}c} {{\text{n}}_{2} + {\text{k}}_{ 2} } \\ {{\text{n}}_{2} } \\ \end{array} } \right)\frac{{\left( {\upalpha^{{{\prime \prime }}} +\upbeta^{{{\prime \prime }}} + {\text{n}}_{2} + 1} \right)_{{{\text{k}}_{ 2} }} }}{{\left( {\upalpha^{{{\prime \prime }}} + 1} \right)_{{{\text{k}}_{ 2} }} }}, $$

we obtain

$$ \int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } $$

$$ .\aleph_{{{\text{P}}_{\text{i}} , {\text{Q}}_{\text{i}} ,\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N}} \left[ {{\text{z}}\left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\;{\text{A}}_{\text{j}} )_{{ 1 , {\text{N}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{a}}_{\text{j}} ,\;{\text{A}}_{\text{j}} ) ]_{{{\text{N}} + 1 , {\text{P}}_{\text{i}} }} }} } \right.} \right] $$

$$ .{\text{P}}_{{{\text{n}}_{1} }}^{{\left( {\upalpha^{{\prime }} ,\upbeta^{{\prime }} } \right)}} \left[ { 1- 2 {\text{z}}_{ 1} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\upmu} } \right] $$

$$ .{\text{P}}_{{{\text{n}}_{2} }}^{{\left( {\upalpha^{{{\prime \prime }}} ,\upbeta^{{{\prime \prime }}} } \right)}} \left[ { 1- 2{\text{z}}_{ 2} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } \right]{\text{dx}} $$

$$ = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} } \right]}} {\sum\limits_{{{\text{k}}_{ 2} = 0}}^{{\left[ {{\text{n}}2} \right]}} {\left( {\begin{array}{*{20}c} {{\text{n}}_{1} +\upalpha^{{\prime }} } \\ {{\text{n}}_{1} - {\text{k}}_{ 1} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {{\text{n}}_{2} +\upalpha^{{{\prime \prime }}} } \\ {{\text{n}}_{2} - {\text{k}}_{ 2} } \\ \end{array} } \right)} } \left( { - {\text{z}}_{ 1} } \right)^{{{\text{k}}_{ 1} }} \left( { - {\text{z}}_{ 2} } \right)^{{{\text{k}}_{ 2} }} $$

$$ .\left( {\begin{array}{*{20}c} {\upalpha^{{\prime }} +\upbeta^{{\prime }} + {\text{n}}_{1} + {\text{k}}_{ 1} } \\ {{\text{k}}_{ 1} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\upalpha^{\prime \prime } +\upbeta^{{{\prime \prime }}} + {\text{n}}_{2} + {\text{k}}_{ 2} } \\ {{\text{k}}_{ 2} } \\ \end{array} } \right) $$

$$ .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} + \left( {\uplambda^{{\prime }} +\upmu^{{\prime }} } \right){\text{k}}_{ 2} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} }} $$

$$ .\aleph_{{{\text{P}}_{\text{i}} { + 2,}\;{\text{Q}}_{\text{i}} + 1 ,\;\uptau_{\text{i}} ; {\text{r}}}}^{{{\text{M,}}\;{\text{N}} + 2}} \left[ {{\text{z}}\left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{b}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{a}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\;{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1,{\text{Q}}_{\text{i}} }} }}^{{\left( { 1\, - \,{\text{u}}\, - \,\uplambda{\text{k}}_{ 1} \, - \,\uplambda^{{\prime }} {\text{k}}_{2} ,\,s} \right),\left( {1\, - \,{\text{v}}\, - \,\upmu{\text{k}}_{ 1} \, - \,\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{t}}} \right)}} } \right.} \right. $$

$$ \left. {\begin{array}{*{20}c} {\left( {{\text{a}}_{\text{j}} ,\;\upalpha_{\text{j}} } \right)_{{ 1 , {\text{N}}}} ,\tau_{j} \left( {{\text{a}}_{\text{j}} ,\;\upalpha_{\text{j}} } \right)_{{{\text{N}} + 1 , {\text{P}}_{i} ;r}} } \\ {\left( {1 - {\text{u}} - {\text{v}} -\uplambda{\text{k}}_{ 1} -\upmu{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} -\upmu^{{\prime }} {\text{k}}_{ 2} ,\;{\text{s}} + {\text{t}}} \right)} \\ \end{array} } \right], $$

(3.2)

valid under the conditions as obtainable from (2.1).

• (C) For the Laguerre polynomials ([9], Eq. (5.1.6), p. 10 and [3], p. 158), we have the following interesting consequence of our result (2.1), by setting

$$ {\text{S}}_{\text{n}}^{ 1} \left[ {\text{x}} \right] \to {\text{L}}_{\text{n}}^{{\left( {\upalpha^{{\prime }} } \right)}} ({\text{x}})\;{\text{in}}\;{\text{which}}\;{\text{case}} $$

$$ {\text{m}}_{1} = 1 ,\quad {\text{A}}_{{{\text{n}}_{1} , {\text{k}}_{ 1} }} = \left( {\begin{array}{*{20}c} {{\text{n}}_{1} +\upalpha^{{\prime }} } \\ {{\text{n}}_{1} } \\ \end{array} } \right)\frac{ 1}{{\left( {\upalpha^{{\prime }} + 1} \right)_{{{\text{k}}_{ 1} }} }} $$

and also taking

$$ {\text{m}}_{2} = 1,\quad {\text{A}}_{{{\text{n}}_{2} ,{\text{k}}_{ 2} }} = \left( {\begin{array}{*{20}c} {{\text{n}}_{2} +\upalpha^{{{\prime \prime }}} } \\ {{\text{n}}_{2} } \\ \end{array} } \right)\frac{ 1}{{\left( {\upalpha^{{{\prime \prime }}} + 1} \right)_{{{\text{k}}_{ 2} }} }}, $$

we get

$$ \int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } $$

$$ .\aleph_{{{\text{P}}_{\text{i}} , {\text{Q}}_{\text{i}} ,\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N}} \left[ {{\text{z}}\left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )_{{ 1 , {\text{N}}}} ,\, \ldots ,\, [\uptau_{\text{i}} ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} ) ]_{{{\text{N}} + 1 , {\text{P}}_{\text{i}} }} }} } \right.} \right] $$

$$ .{\text{L}}_{{{\text{n}}_{1} }}^{{\left( {\upalpha^{{\prime }} } \right)}} \left[ {{\text{z}}_{ 1} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\upmu} } \right]\quad .{\text{L}}_{{{\text{n}}_{2} }}^{{\left( {\upalpha^{{{\prime \prime }}} } \right)}} \left[ {{\text{z}}_{ 2} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } \right]{\text{dx}} $$

$$ = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} } \right]}} {\sum\limits_{{{\text{k}}_{ 2} = 0}}^{{\left[ {{\text{n}}_{2} } \right]}} {\frac{{\left( { - {\text{n}}_{1} } \right)_{{{\text{k}}_{ 1} }} \left( { - {\text{n}}_{2} } \right)_{{{\text{k}}_{2} }} }}{{{\text{k}}_{ 1} ! {\text{k}}_{ 2} !}}\left( {\begin{array}{*{20}c} {{\text{n}}_{1} +\upalpha^{{\prime }} } \\ {{\text{n}}_{1} } \\ \end{array} } \right)\frac{1}{{\left( {\upalpha^{{\prime }} + 1} \right){\text{k}}_{ 1} }}\left( {\begin{array}{*{20}c} {{\text{n}}_{2} +\upalpha^{{{\prime \prime }}} } \\ {{\text{n}}_{2} } \\ \end{array} } \right)\frac{1}{{\left( {\upalpha^{\prime \prime } + 1} \right){\text{K}}_{ 2} }}} } {\text{z}}_{1}^{{{\text{k}}_{ 1} }} {\text{z}}_{2}^{{{\text{k}}_{ 2} }} $$

$$ .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} + \left( {\uplambda^{{\prime }} +\upmu^{{\prime }} } \right){\text{k}}_{ 2} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} }} $$

$$ \begin{array}{*{20}l} {.\aleph_{{{\text{P}}_{\text{i}} {\text{ + 2,Q}}_{\text{i}} { + 1,}\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N + 2}} \left[ {{\text{z}}\left( {\frac{{{\text{b}}\, - \,{\text{a}}}}{{{\text{b}}\, - \,{\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}}\, - \,{\text{a}}}}{{{\text{a}}\, - \,{\text{c}}}}} \right)^{\text{t}} \left| \begin{aligned} & \left( { 1- {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{2} ,s} \right),\,\left( {1 - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} ,{\text{t}}} \right) \\ & ( {\text{b}}_{\text{j}} , {\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\ldots , [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} , {\text{B}}_{\text{j}} ) ]_{{{\text{M + 1,Q}}_{\text{i}} }} \\ \end{aligned} \right.} \right.} \hfill \\ {\left. {\begin{array}{*{20}c} {\left( {{\text{a}}_{\text{j}} ,\,\upalpha_{\text{j}} } \right)_{{ 1 , {\text{N}}}} ,\,\tau_{j} \left( {{\text{a}}_{\text{j}} ,\,\upalpha_{\text{j}} } \right)_{{{\text{N}} + 1 , {\text{P}}_{i} ;r}} } \\ {\left( {1 - {\text{u}} - {\text{v}} -\uplambda{\text{k}}_{ 1} -\upmu{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} -\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{s}} + {\text{t}}} \right)} \\ \end{array} } \right],} \hfill \\ \end{array} $$

(3.3)

suitable under the conditions as required sufficiently for (2.1).

• (D) Letting ${\text{n}}_{2}$  → 0 in (2.1), we have

$$ \int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } $$

$$ .\aleph_{{{\text{P}}_{\text{i}} , {\text{Q}}_{\text{i}} ,\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N}} \left[ {{\text{z}}\left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{ 1,{\text{M}}}} ,\ldots , [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1 , {\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )_{{ 1,{\text{N}}}} ,\ldots , [\uptau_{\text{i}} ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} ) ]_{{{\text{N}} + 1 , {\text{P}}_{\text{i}} }} }} } \right.} \right] $$

$$ .{\text{S}}_{{{\text{n}}_{1} }}^{{{\text{m}}_{1} }} \left[ {{\text{z}}_{ 1} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\upmu} } \right]{\text{dx}} $$

$$ = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} /{\text{m}}_{1} } \right]}} {\frac{{\left( { - {\text{n}}_{1} } \right)_{{{\text{m}}_{1} {\text{k}}_{ 1} }} }}{{{\text{k}}_{1} !}}} {\text{A}}_{{{\text{n}}_{1} , {\text{k}}_{ 1} }} {\text{z}}_{ 1}^{{{\text{k}}_{ 1} }} .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} }} $$

$$ \begin{array}{*{20}l} {.\aleph_{{{\text{P}}_{\text{i}} {\text{ + 2,Q}}_{\text{i}} { + 1,}\uptau_{\text{i}} ; {\text{r}}}}^{\text{M,N + 2}} \left[ {{\text{z}}\left( {\frac{{{\text{b}}\, - \,{\text{a}}}}{{{\text{b}}\, - \,{\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}}\, - \,{\text{a}}}}{{{\text{a}}\, - \,{\text{c}}}}} \right)^{\text{t}} \left| \begin{aligned} & \left( { 1- {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{2} ,s} \right),\left( {1 - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} ,{\text{t}}} \right) \\ & ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} ,\ldots , [\uptau_{\text{i}} ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} ) ]_{{{\text{M + 1,Q}}_{\text{i}} }} \\ \end{aligned} \right. \, } \right.} \hfill \\ {\left. {\begin{array}{*{20}c} {\left( {{\text{a}}_{\text{j}} ,\upalpha_{\text{j}} } \right)_{{ 1 , {\text{N}}}} ,\tau_{j} \left( {{\text{a}}_{\text{j}} ,\upalpha_{\text{j}} } \right)_{{{\text{N}} + 1 , {\text{P}}_{i} ;r}} } \\ {\left( {1 - {\text{u}} - {\text{v}} -\uplambda{\text{k}}_{ 1} -\upmu{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} -\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{s}} + {\text{t}}} \right)} \\ \end{array} } \right],} \hfill \\ \end{array} $$

(3.4)

valid under the conditions as essential for (2.1).

• (E) Taking $\tau_{i} \to 1$ in (2.1), the I-function given by Saxena [2, 3] is obtained from Aleph function and the main integral (2.1) converts in the following form:

$$ \begin{array}{*{20}l} {\int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } } \hfill \\ {.{\text{I}}_{{{\text{P}}_{\text{i}} , {\text{Q}}_{\text{i}} ; {\text{r}}}}^{\text{M,N}} \left[ {{\text{z}}\left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{ 1 , {\text{M}}}} , ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{{\text{M}} + 1,{\text{Q}}_{\text{i}} }} }}^{{ ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )_{{ 1 , {\text{N}}}} , ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )_{{{\text{N}} + 1 , {\text{P}}_{\text{i}} }} }} } \right.} \right]} \hfill \\ \end{array} $$

$$ .{\text{S}}_{{{\text{n}}_{1} }}^{{{\text{m}}_{1} }} \left[ {{\text{z}}_{ 1} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\upmu} } \right]{\text{S}}_{{{\text{n}}_{2} }}^{{{\text{m}}_{2} }} \left[ {{\text{z}}_{ 2} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } \right]{\text{dx}} $$

$$ = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} /{\text{m}}_{1} } \right]}} {\sum\limits_{{{\text{k}}_{ 2} = 0}}^{{\left[ {{\text{n}}_{2} /{\text{m}}_{2} } \right]}} {\frac{{\left( { - {\text{n}}_{1} } \right)_{{{\text{m}}_{1} {\text{k}}_{ 2} }} \left( { - {\text{n}}_{2} } \right)_{{{\text{m}}_{2} {\text{k}}_{ 2} }} }}{{{\text{k}}_{1} !{\text{k}}_{2} !}}} } {\text{A}}_{{{\text{n}}_{1} ,{\text{k}}_{ 1} }} {\text{A}}_{{{\text{n}}_{2} ,{\text{k}}_{ 2} }} {\text{z}}_{ 1}^{{{\text{k}}_{ 1} }} {\text{z}}_{ 2}^{{{\text{k}}_{ 2} }} $$

$$ .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} + \left( {\uplambda^{{\prime }} +\upmu^{{\prime }} } \right){\text{k}}_{ 2} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} }} $$

$$ .{\text{I}}_{{{\text{P}}_{\text{i}} {\text{ + 2,Q}}_{\text{i}} {\text{ + 1;r}}}}^{{{\text{M,N}} + 2}} \left[ {{\text{z}}\left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{b}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{a}}}}{{{\text{a}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )_{{ 1,{\text{M}}}} ,\, ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} ) ]_{{{\text{M}} + 1,{\text{Q}}_{\text{i}} }} }}^{{\left( { 1\, - \,{\text{u}}\, - \,\uplambda{\text{k}}_{ 1} \, - \,\uplambda^{{\prime }} {\text{k}}_{2} ,\,s} \right),\left( {1\, - \,{\text{v}}\, - \,\upmu{\text{K}}_{ 1} \, - \,\upmu^{{\prime }} {\text{K}}_{ 2} ,\,{\text{t}}} \right)}} } \right.} \right. $$

$$ \left. {\begin{array}{*{20}c} {\left( {{\text{a}}_{\text{j}} ,\upalpha_{\text{j}} } \right)_{{ 1 , {\text{N}}}} ,\left( {{\text{a}}_{\text{j}} ,\upalpha_{\text{j}} } \right)_{{{\text{N}} + 1 , {\text{P}}_{i} ;r}} } \\ {\left( {1\, - \,{\text{u}}\, - \,{\text{v}}\, - \,\uplambda{\text{k}}_{ 1} \, - \,\upmu{\text{k}}_{ 1} \, - \,\uplambda^{{\prime }} {\text{k}}_{ 2} \, - \,\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{s}} + {\text{t}}} \right)} \\ \end{array} } \right], $$

(3.5)

valid under the conditions as required sufficiently for (2.1).

• (F) If we take $\tau_{i} \to 1$ and r = 1 in (2.1), the Aleph function reduces to Fox’s H-function [1] and the main integral takes the following form:

$$ \int\limits_{\text{a}}^{\text{b}} {\left( {{\text{x}} - {\text{a}}} \right)^{{{\text{u}} - 1}} \left( {{\text{b}} - {\text{x}}} \right)^{{{\text{v}} - 1}} \left( {{\text{x}} - {\text{c}}} \right)^{{ - {\text{u}} - {\text{v}}}} } \quad .{\text{H}}_{{{\text{P}},{\text{Q}}}}^{{{\text{M}},{\text{N}}}} \left[ {{\text{z}}\left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{\text{j}} ,\,{\text{B}}_{\text{j}} )}}^{{ ( {\text{a}}_{\text{j}} ,\,{\text{A}}_{\text{j}} )}} } \right.} \right] $$

$$ .{\text{S}}_{{{\text{n}}_{1} }}^{{{\text{m}}_{1} }} \left[ {{\text{z}}_{ 1} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\uplambda} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{\upmu} } \right]{\text{S}}_{{{\text{n}}_{2} }}^{{{\text{m}}_{2} }} \left[ {{\text{z}}_{ 2} \left( {\frac{{{\text{x}} - {\text{a}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\uplambda^{{\prime }} }} \left( {\frac{{{\text{b}} - {\text{x}}}}{{{\text{x}} - {\text{c}}}}} \right)^{{\upmu^{{\prime }} }} } \right]{\text{dx}} $$

$$ = \sum\limits_{{{\text{k}}_{1} = 0}}^{{\left[ {{\text{n}}_{1} /{\text{m}}_{1} } \right]}} {\sum\limits_{{{\text{k}}_{ 2} = 0}}^{{\left[ {{\text{n}}_{2} /{\text{m}}_{2} } \right]}} {\frac{{\left( { - {\text{n}}_{1} } \right)_{{{\text{m}}_{1} {\text{k}}_{ 2} }} \left( { - {\text{n}}_{2} } \right)_{{{\text{m}}_{2} {\text{k}}_{ 2} }} }}{{{\text{k}}_{1} !{\text{k}}_{2} !}}} } {\text{A}}_{{{\text{n}}_{1} , {\text{k}}_{ 1} }} {\text{A}}_{{{\text{n}}_{2} , {\text{k}}_{ 2} }} {\text{z}}_{ 1}^{{{\text{k}}_{ 1} }} {\text{z}}_{ 2}^{{{\text{k}}_{ 2} }} $$

$$ .\left( {{\text{b}} - {\text{a}}} \right)^{{{\text{u}} + {\text{v}} + \left( {\uplambda +\upmu} \right){\text{k}}_{ 1} + \left( {\uplambda^{{\prime }} +\upmu^{{\prime }} } \right){\text{k}}_{ 2} - 1}} \left( {{\text{b}} - {\text{c}}} \right)^{{ - {\text{u}} -\uplambda{\text{k}}_{ 1} -\uplambda^{{\prime }} {\text{k}}_{ 2} }} \left( {{\text{a}} - {\text{c}}} \right)^{{ - {\text{v}} -\upmu{\text{k}}_{ 1} -\upmu^{{\prime }} {\text{k}}_{ 2} }} $$

$$ .{\text{H}}_{{{\text{P}} + 2,{\text{Q}} + 1,}}^{{{\text{M}},{\text{N}} + 2}} \left[ {{\text{z}}\left( {\frac{{{\text{b}}\, - \,{\text{a}}}}{{{\text{b}}\, - \,{\text{c}}}}} \right)^{\text{s}} \left( {\frac{{{\text{b}}\, - \,{\text{a}}}}{{{\text{a}}\, - \,{\text{c}}}}} \right)^{\text{t}} \, \left| {_{{ ( {\text{b}}_{ 1} ,\,{\text{B}}_{ 1} ),\, ( {\text{b}}_{\text{q}} ,\,{\text{B}}_{\text{q}} )\left( {1\, - \,{\text{u}}\, - \,{\text{v}}\, - \,\uplambda{\text{k}}_{ 1} \, - \,\upmu{\text{K}}_{ 1} \, - \,\uplambda^{{\prime }} {\text{k}}_{ 2} \, - \,\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{s}} + {\text{t}}} \right)}}^{{\left( { 1\, - \,{\text{u}}\, - \,\uplambda{\text{k}}_{ 1} \, - \,\uplambda^{{\prime }} {\text{k}}_{2} ,\,s} \right),\,\left( {1\, - \,{\text{v}}\, - \,\upmu{\text{k}}_{ 1} \, - \,\upmu^{{\prime }} {\text{k}}_{ 2} ,\,{\text{t}}} \right)\left( {{\text{a}}_{ 1} ,\,\upalpha_{ 1} } \right),\,\left( {{\text{a}}_{\text{p}} ,\,\upalpha_{\text{p}} } \right)}} } \right.} \right], $$

(3.6)

valid under the conditions as required sufficiently for (2.1).

The significance of outcomes lies in its various generalizations. In perspective of the generality of the function and polynomials of very broad nature involved in the results, our results encompass several particular cases of interest scattered hitherto in the literature.