Investigation of some finite integrals incorporating incomplete Aleph functions

1. Introduction

In the discipline of mathematics, special functions are vital. From time to time modifications in the approach and concepts of these functions led us to the latest form of generalized functions. Many mathematicians have done lots of remarkable work which has opened new aspects of research as well.^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}

Kumar et al.²² derive various integral formulas involving Aleph-function multiplied with algebraic functions and special functions. After that, Bansal et al.²³ deal with several finite and infinite integrals involving the family of incomplete H-functions. Furthermore, points out some known and new special cases of these integrals. Finally, we establish the integral representation of incomplete H-functions. Multivariable Aleph functions with algebraic functions, Jacobi polynomials, Legendre functions, Bessel–Maitland functions, and general class of polynomials integrals evaluated by Tadesse et al.²⁴ Some integral formulae of incomplete I-functions involving the exponential function, the Legendre polynomials and generalized Laguerre polynomials are given by Bhatter et al.²⁵ and Kumar et al.²⁶ defined general definite integrals involving the product of the Aleph function and generalized incomplete hypergeometric function with general arguments.

Here, we use a finite integral operator to derive integral formulas that combine incomplete Aleph functions with algebraic and other well-known special functions. This allows us to classify special functions and integral formulas into distinct classes with similar properties. Furthermore, this framework empowers us to create novel integral formulas that find practical applications in a wide range of areas, including science, engineering, probability distribution, electro-optics, non-linear wave propagation, electromagnetism, potential theory, electric circuits, quantum mechanics, and numerous other domains.

2. Pre-Requisites

In this section, we furnish significant and pivotal definitions for the functions that are employed and play a crucial role in the context of this study.

2.1. Gamma function

The gamma function is most straightforwardly understood as an extension of the factorial concept to encompass all real numbers. It’s defined by

2.2. Beta function

The beta function $B(\alpha ,\beta ),$ often known as Euler’s integral of first kind, plays a pivotal role in factorial calculus. In terms of the gamma function, the Beta function is expressed as follows:

2.3. Incomplete gamma functions

Prym,²⁷ introduced the incomplete gamma functions (IGFs) $\gamma (S,Y)$ and $\mathrm{\Gamma }(S,Y)$ as follows :

and

$$\mathrm{\Gamma }(S,Y)={\int }_Y^{\infty }{T}^{S-1}{e}^{-T}dT,(Y\ge 0;ℝ(S)>0\mathrm{\text{when}}Y=0).$$(2.4)

Here, $\gamma (S,Y)+\mathrm{\Gamma }(S,Y)=\mathrm{\Gamma }(S),$ where $\mathrm{\Gamma }(S)$ is the well-known gamma function. This property is called the decomposition formula.

2.4. Incomplete Aleph functions

The Incomplete Aleph functions (IAFs) ${}^{(\gamma )}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}(Z)$ and ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}(Z)$ were introduced by Bansal et al.²⁸ and are given as

where $z\ne 0$ and and

$${}^{(\gamma )}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}\left[Z\left|\begin{array}{c}({𝔪}_1,{𝔄}_1,Y),{({𝔪}_j,{𝔄}_j)}_{2,N},\hfill \\ {[{\rho }_j({𝔪}_{ji},{𝔄}_{ji})]}_{N+1,P_i}\hfill \\ {({𝔫}_j,{𝔅}_j)}_{1,M},{[{\rho }_j({𝔫}_{ji},{𝔅}_{ji})]}_{M+1,Q_i}\hfill \end{array}\right]\right.=\frac{1}{2\pi i}{\int }_L\varphi (\zeta ,Y)Z^{-\zeta }d\zeta ,$$(2.7)

where $z\ne 0$ and The Incomplete Aleph functions ${}^{(\gamma )}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}(Z)$ and ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}(Z)$ exist for all $Y\ge 0$ and for more conditions (see Refs. 28,29 and 30).

The IAFs ${}^{(\gamma )}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}(Z)$ and ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}(Z)$ given in (2.7) and (2.5) convert to many known special functions by substituting the specific values of parameters named as the $\aleph$-function introduced by Südland,³¹ the incomplete I-functions defined by Bansal and Kumar,^32,33 the I-function introduced by Saxena,³⁴ the incomplete H-functions introduced by Srivastava,³⁵ the familiar Fox’s H-function.³⁶

This paper has five sections. Section 1 introduces Special functions. Section 2 lists the prerequisites for reading this paper. Section 3 discusses the main results and provides the necessary theorems with proofs. Section 4 presents some special cases of results, and Sec. 5 concludes the paper. Finally, the authors acknowledge the researchers and mathematicians whose work helped complete this paper.

3. Main Results

In this section, we evaluate several finite integrations associated with the incomplete Aleph functions and some special types of functions.

3.1. Finite integrals associate the algebraic functions and incomplete Aleph functions

Here, we are conducting an evaluation or analysis of the incomplete Aleph functions, specifically concerning their relationship or interactions with certain algebraic functions.

Theorem 3.1.

Proof. We can express the left-hand side of (3.1) using a Mellin–Barnes integral, and with some simplification, we obtain

the above equation can be expressed as follows:

$$=\frac{1}{2\pi i}{\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta ){\prod }_{j=2}^N\hfill \\ \times \mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta ){\prod }_{j=N+1}^{P_i}\hfill \\ \times \mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\times \frac{\mathrm{\Gamma }(1-𝔱-\zeta )\mathrm{\Gamma }(𝔱-\mathcal{𝒦})}{\mathrm{\Gamma }(1-\mathcal{𝒦}-\zeta )}{Z}^{-\zeta }d\zeta ,$$

after a minor simplification, we achieve the desired outcome. □

Theorem 3.2.

Proof. To prove this theorem, follow the identical steps as those in Theorem 3.1. □

Theorem 3.3.

Proof. Use the Mellin–Barnes integral form of incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ given in (2.6) on left-hand side of (3.3) and after a bit of simplification, we have

we can write the above equation as follows:

$$=\frac{1}{2\pi i}{\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta ){\prod }_{j=2}^N\hfill \\ \times \mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta ){\prod }_{j=N+1}^{P_i}\hfill \\ \times \mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\times \frac{\mathrm{\Gamma }(𝔱-\zeta )\mathrm{\Gamma }(\mathcal{𝒦})}{\mathrm{\Gamma }(𝔱+\mathcal{𝒦}-\zeta )}{Z}^{-\zeta }d\zeta ,$$

after a small simplification, we get the desired result. □

Theorem 3.4.

Proof. Do the same as Theorem 3.3. □

Theorem 3.5.

Proof. On the left-hand side of (3.5) transform incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ in Mellin–Barnes integral form (2.6), and after a brief simplification, we obtain

substituting $𝔱=𝔎+1\Rightarrow d𝔱=d𝔎$ and using formula $\mathrm{\Gamma }(c)\mathrm{\Gamma }(e)=\mathrm{\Gamma }(c+e){\int }_0^{\infty }{x}^{c-1}{(x+1)}^{-(c+e)}dx=\mathrm{\Gamma }(c+e){\int }_0^{\infty }{x}^{e-1}{(x+1)}^{-(c+e)}dx.$ We get

$$=\frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)Z^{-\zeta }\times \left[{\int }_0^{\infty }{𝔎}^{\mathcal{𝒦}-1}{(1+𝔎)}^{-(\mathcal{𝒦}+𝔱+\zeta -\mathcal{𝒦})}d𝔎\right]d\zeta ,$$

we can write the above equation as

$$=\frac{1}{2\pi i}{\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta ){\prod }_{j=2}^N\hfill \\ \times \mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta ){\prod }_{j=N+1}^{P_i}\hfill \\ \times \mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\times \frac{\mathrm{\Gamma }(𝔱+\zeta -\mathcal{𝒦})\mathrm{\Gamma }(\mathcal{𝒦})}{\mathrm{\Gamma }(𝔱+\zeta )}{Z}^{-\zeta }d\zeta ,$$

after a small simplification, we get the desired result. □

Theorem 3.6.

Proof. Do the same as Theorem 3.5. □

Theorem 3.7.

Proof. Use the Mellin–Barnes integral form of incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ given in (2.6) on left-hand side of (3.7) and after a bit of simplification, we have

substituting $\delta =𝔎b\Rightarrow d\delta =bd𝔎.$ We get

$$=\frac{1}{2\pi i}{b}^{𝔱-\mathcal{𝒦}}{\int }_L\phi (\zeta ,Y){(Zb)}^{-\zeta }\times \left[{\int }_0^{\infty }{𝔎}^{𝔱-\zeta -1}{(1+𝔎)}^{-(\mathcal{𝒦}-𝔱+\zeta +𝔱-\zeta )}d𝔎\right]d\zeta ,$$

we can write the above equation as

$$=\frac{1}{2\pi i}{b}^{-𝔱+\mathcal{𝒦}}{\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta )\hfill \\ \times {\prod }_{j=2}^N\mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta )\hfill \\ \times {\prod }_{j=N+1}^{P_i}\mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\times \frac{\mathrm{\Gamma }(𝔱-\zeta )\mathrm{\Gamma }(\mathcal{𝒦}-𝔱+\zeta )}{\mathrm{\Gamma }(\mathcal{𝒦})}{(Zb)}^{-\zeta }d\zeta ,$$

we obtain the intended outcome by making a modest simplification. □

Theorem 3.8.

Proof. To establish the validity of this theorem, follow the identical steps as outlined in Theorem 3.7. □

Theorem 3.9.

Proof. On the left-hand side of (3.9) transform incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ in Mellin–Barnes integral form (2.6), and after a brief simplification, we obtain

By using the formula given in Ref. 37, p. 261 as follows:

$${\int }_{-1}^1{(1-y)}^{m+a}{(1+y)}^{m+b}dy=2^{2m+a+b+1}B(1+a+m,1+b+m),$$(3.10)

we can write the above equation as

$$=\frac{1}{2\pi i}{2}^{𝔱+\mathcal{𝒦}+1}{\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta )\hfill \\ \times {\prod }_{j=2}^N\mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta )\hfill \\ \times {\prod }_{j=N+1}^{P_i}\mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\times \frac{\mathrm{\Gamma }(1+\mathcal{𝒦})\mathrm{\Gamma }(1+𝔱-\nu \zeta )}{\mathrm{\Gamma }(2+\mathcal{𝒦}+𝔱-\nu \zeta )}{(2^{\nu }Z)}^{-\zeta }d\zeta ,$$

we get the intended outcome by making a modest change. □

Theorem 3.10.

Proof. Follow the same steps as followed in Theorem 3.9. □

3.2. Finite integrals associate the Jacobi polynomials and incomplete Aleph functions

Jacobi polynomial $P_m^{(\alpha ,\beta )}(x)$³⁷ is given as

where ${}_2{F}_1$ is a well-known hyper-geometric function. For $\alpha =0,\beta =0,$ Jacobi polynomial (3.12) converts to Legendre polynomial (see Ref. 37, p. 257). For $y=1$, $P_m^{(\alpha ,\beta )}(y)=\frac{{(1+\alpha )}_m}{m!}.$

This section is dedicated to deriving integral formulas that incorporate incomplete Alpha functions with Jacobi polynomials.

Theorem 3.11. If $\alpha >-1,\beta >-1,ℝ(\mathrm{\Lambda })>-1$ and $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ then a integral representation with the incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ and the Jacobi polynomials is as follows:

where $A=(\ell -\beta -k,\mu ),(\ell -k,\mu ),({𝔪}_1,{𝔄}_1,Y),{({𝔪}_j,{𝔄}_j)}_{2,N},{[{\rho }_j({𝔪}_{ji},{𝔄}_{ji})]}_{N+1,P_i}$ and $B={({𝔫}_j,{𝔅}_j)}_{1,M},(-\beta -\ell -n-k,\mu ),(-1-\ell -\alpha -n-k,\mu ),{[{\rho }_j({𝔫}_{ji},{𝔅}_{ji})]}_{M+1,Q_i}.$

Proof. Use the Mellin–Barnes integral representation of incomplete Alpeh function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ on the left-hand side of (3.13) and after a bit of simplification, we have

Now using the formula

$${\int }_{-1}^1{\delta }^{\mathrm{\Lambda }}{(1-\delta )}^{\alpha }{(1+\delta )}^{\ell }{P}_n^{(\alpha ,\beta )}(\delta )d\delta ={(-1)}^n\frac{2^{\alpha +\ell +1}\mathrm{\Gamma }(\alpha +n+1)\mathrm{\Gamma }(\ell +1)\mathrm{\Gamma }(\beta +\ell +1)}{n!\mathrm{\Gamma }(\beta +\ell +n+1)\mathrm{\Gamma }(\alpha +\ell +n+2)}\times {}_3{F}_2\left[\begin{array}{c}-\mathrm{\Lambda },\ell +\beta +1,\ell +1;\hfill \\ \ell +\beta +n+1,\ell +\alpha +n+2;\hfill \end{array}1\right],$$(3.14)

here ${}_3{F}_2$ is a special case of generalized hypergeometric function.

Hence

$$=\frac{{(-1)}^n{2}^{\alpha +\ell +1}\mathrm{\Gamma }(\alpha +n+1)}{n!}\sum _{k=0}^{\infty }\frac{{(-\mathrm{\Lambda })}_k{(1)}^k}{k!}\frac{1}{2\pi i}\times {\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta ){\prod }_{j=2}^N\hfill \\ \times \mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta ){\prod }_{j=N+1}^{P_i}\hfill \\ \times \mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\times \frac{\mathrm{\Gamma }(\ell -\mu \zeta +\beta +k+1)\mathrm{\Gamma }(\ell -\mu \zeta +k+1)}{\begin{array}{c}\mathrm{\Gamma }(\ell -\mu \zeta +\beta +n+k+1)\hfill \\ \times \mathrm{\Gamma }(\ell -\mu \zeta +\alpha +n+k+2)\hfill \end{array}}{(2^{\mu }Z)}^{\zeta },$$

we get the intended outcome by making a modest change. □

Theorem 3.12. If $\alpha >-1$, $\beta >-1$, $ℝ(\mathrm{\Lambda })>-1$ and $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ then an integral representation with the incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{P_i,Q_i,{\rho }_i;R}^{M,N}$ and the Jacobi polynomials is as follows:

where $A=(\ell -\beta -k,\mu ),(\ell -k,\mu ),({𝔪}_1,{𝔄}_1,Y),{({𝔪}_j,{𝔄}_j)}_{2,N},{[{\rho }_j({𝔪}_{ji},{𝔄}_{ji})]}_{N+1,P_i}$ and $B={({𝔫}_j,{𝔅}_j)}_{1,M},(-\beta -\ell -n-k,\mu ),(-1-\ell -\alpha -n-k,\mu ),{[{\rho }_j({𝔫}_{ji},{𝔅}_{ji})]}_{M+1,Q_i}.$

Proof. Do this as done in Theorem 4.1. □

Theorem 3.13.

provided $\alpha >0,ℝ(\ell )>-1$ and $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}.$

Proof. The left-hand side of (3.16) can be written as

Now using the formula given in (3.12), we obtain

$$=\frac{{(1+\omega )}_m}{m!}\sum _{k=0}^{\infty }\frac{{(1+\omega +\tau +m)}_k{(-m)}_k}{{(1+\omega )}_k{2}^{k}k!}\times \frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)Z^{-\zeta }\times \left[{\int }_{-1}^1{(1-\delta )}^{\alpha -\mu \zeta +k}{(1+\delta )}^{\ell }(\delta )P_n^{(\omega ,\tau )}(\delta )d\delta \right]d\zeta .$$(3.17)

Again using (3.12) in (3.17), we have

$$\begin{array}{c}=\frac{\mathrm{\Gamma }\mathrm{\Gamma }(1+\kappa +n)(1+\omega +m)}{\mathrm{\Gamma }(1+\kappa )m!n!}\times \sum _{k=0}^{\infty }\frac{{(1+\omega +\tau +m)}_k{(1+\kappa +\ell +n)}_k{(-m)}_k{(-n)}_k}{\mathrm{\Gamma }(1+\omega +k)\mathrm{\Gamma }(1+\kappa +k)2^{2k}{(k!)}^2}\times \frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)Z^{-\zeta }{2}^{-\mu \zeta }\hfill \\ \times \left[{\int }_{-1}^1{(1-\delta )}^{\alpha -\mu \zeta +2k}{(1+\delta )}^{\ell }d\delta \right]d\zeta .\hfill \end{array}$$(3.18)

using (3.10) in (3.18), we get

$$\begin{array}{c}=2^{\alpha +\ell +1}\frac{\mathrm{\Gamma }(1+\kappa +n)\mathrm{\Gamma }(1+\omega +m)}{m!n!}\times \sum _{k=0}^{\infty }\frac{{(1+\kappa +\ell +n)}_k{(1+\omega +\tau +m)}_k{(-m)}_k{(-n)}_k}{\mathrm{\Gamma }(1+\omega +k)\mathrm{\Gamma }(1+\kappa +k){(k!)}^2}\times \frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)Z^{-\zeta }{2}^{-\mu \zeta }\hfill \\ \times \left[\frac{\mathrm{\Gamma }(1+\alpha -\mu \zeta +2k)\mathrm{\Gamma }(1+\ell )}{\mathrm{\Gamma }(2+\alpha -\mu \zeta +2k+\ell )}\right]d\zeta ,\hfill \end{array}$$

we get the intended outcome by making a modest change. □

Theorem 3.14.

provided $\alpha >0,ℝ(\ell )>-1$ and $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}.$

Proof. Do this as done in Theorem 4.3. □

Theorem 3.15. If $ℝ(\kappa )>-1$, $ℝ(\ell )>-1$ and $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ then the following integral equation holds:

where $C=(-\omega -k,\mu ),(-\tau ,\beta ),({𝔪}_1,{𝔄}_1,Y),{({𝔪}_j,{𝔄}_j)}_{2,N},{[{\rho }_j({𝔪}_{ji},{𝔄}_{ji})]}_{N+1,P_i}$ and $D={({𝔫}_j,{𝔅}_j)}_{1,M},(-1-\omega -\tau -k,\mu +\beta ),{[{\rho }_j({𝔫}_{ji},{𝔅}_{ji})]}_{M+1,Q_i}$.

Proof. The left-hand side of (3.20) can be written as a Mellin–Barnes integral. After some simplification, this becomes

By using (3.10) in (3.21), we arrive at

$$\begin{array}{c}=2^{\omega +\tau +1}\frac{{(1+\kappa )}_n}{n!}\sum _{k=0}^{\infty }\frac{{(1+\kappa +\ell +n)}_k{(-n)}_k}{{(1+\kappa )}_{k}k!}\times \frac{1}{2\pi i}{\int }_L\frac{\begin{array}{c}\mathrm{\Gamma }(1-{𝔪}_1-{𝔄}_1\zeta ,Y){\prod }_{j=1}^M\mathrm{\Gamma }({𝔫}_j+{𝔅}_j\zeta )\hfill \\ \times {\prod }_{j=2}^N\mathrm{\Gamma }(1-{𝔪}_j-{𝔄}_j\zeta )\hfill \end{array}}{\begin{array}{c}{\sum }_{i=1}^R{\rho }_i[{\prod }_{j=M+1}^{Q_i}\mathrm{\Gamma }(1-{𝔫}_{ji}-{𝔅}_{ji}\zeta )\hfill \\ \times {\prod }_{j=N+1}^{P_i}\mathrm{\Gamma }({𝔪}_{ji}+{𝔄}_{ji}\zeta )]\hfill \end{array}}\hfill \\ \times \left[\frac{\mathrm{\Gamma }(1+\omega -\mu \zeta +k)\mathrm{\Gamma }(1+\tau -\beta \zeta )}{\mathrm{\Gamma }(2+\omega +\tau -\mu \zeta -\beta \zeta +k)}\right]{(2^{-\mu +\beta }Z)}^{\zeta }d\zeta .\hfill \end{array}$$(3.22)

After a few simple steps, we arrive at the desired result. □

Theorem 3.16.

provided $ℝ(\kappa )>-1$, $ℝ(\ell )>-1$ and $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}.$

Proof. To prove this theorem, use the same approach as in the proof of Theorem 4.5. □

Theorem 3.17.

provided $ℝ(c)>-1$, $ℝ(d)>-1$, $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ and $ℝ[\omega +\mu min(\frac{{𝔢}_{𝔧}}{d_j})]>-1(j=\overline{1,M}).$

Proof. We can express the left-hand side of (3.24) in the form of Mellin–Barnes integral, and following some simplification, we obtain

By using (3.10) in (3.25), we arrive at

$$=\frac{2^{\omega +\tau +1}{(1+c)}_n}{n!}\times \sum _{k=0}^{\infty }\frac{\mathrm{\Gamma }(1+\omega +k){(1+c+d+n)}_k{(-n)}_k}{{(1+c)}_{k}k!}\times \frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)Z^{-\zeta }\left[\frac{\mathrm{\Gamma }(1+\omega +k)\mathrm{\Gamma }(1+\tau +\mu \zeta )}{\mathrm{\Gamma }(2+\omega +\tau +\mu \zeta +k)}\right]\times {(2^{-\mu }Z)}^{-\zeta }d\zeta .$$

We can simplify the equation and arrive at the desired result. □

Theorem 3.18.

provided $ℝ(c)>-1$, $ℝ(d)>-1$, $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ and $ℝ[\omega +\mu min(\frac{{𝔢}_{𝔧}}{d_j})]>-1(j=\overline{1,M}).$

Proof. Do the same as in Theorem 4.7. □

Theorem 3.19.

provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$, $ℝ[\omega +\mu min(\frac{{𝔢}_{𝔧}}{d_j})]>-1$ and $ℝ[\tau +\beta min(\frac{{𝔢}_{𝔧}}{d_j})]>-1(j=\overline{1,M}).$

Proof. To prove this theorem follow the same steps as in Theorem 4.7. □

3.3. Finite integrals associate the Legendre function and incomplete Aleph functions

The Legendre’s differential equation (see Sec 3.1 of Ref. 38) can be written as

where $x,a,b$ are unrestricted. It is the Legendre function.

If we put $f={(x^2-1)}^{1∕2b}a,$ we can write (3.28) as

and with $D=1∕2-z∕2$ as an independent variable, the above equation reduces to Solution of Legendre’s differential equation (3.28) in terms of Gauss hypergeometric equation with $\alpha =b-a$, $\beta =b+a+1$ and $\gamma =b+1,$ is as follows:

$$f=P_a^b(x)=\frac{1}{\mathrm{\Gamma }(1-b)}{\left(\frac{x+1}{x-1}\right)}^{1∕2b}\times {}_2{F}_1[-a,a+1;1-b;1∕2-x∕2],|1-x|<2,$$

where Legendre function of first kind³⁸ representing by $P_a^b(x)$.

Now, evaluate the main findings of this section by using Legendre functions.

Theorem 3.20.

provided $\tau >0$, $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ and $\kappa \in \mathbb{N}\cup 0.$ where $A=(1-\tau ,\omega ),({𝔪}_1,{𝔄}_1,Y),{({𝔪}_j,{𝔄}_j)}_{2,N},{[{\rho }_j({𝔪}_{ji},{𝔄}_{ji})]}_{N+1,P_i}$ and $B={({𝔫}_j,{𝔅}_j)}_{1,M},(1∕2-\kappa ∕2+\ell ∕2-\tau ∕2,\omega ∕2),(-\kappa ∕2-\ell ∕2-\tau ∕2,\omega ∕2),{[{\rho }_j({𝔫}_{ji},{𝔅}_{ji})]}_{M+1,Q_i}.$

Proof. The left-hand side of (3.29) can be expressed as Mellin–Barnes integral. After some simplification, we obtain

Now, using formula (see Sec 3.12 of Ref. 38) for $ℝ(\tau )>0$, $\kappa \in \mathbb{N}$ as follows:

$${\int }_0^1{\delta }^{\tau }{(1-{\delta }^2)}^{\kappa ∕2}{P}_{\ell }^{\kappa }(\delta )d\delta =\frac{{(-1)}^{\kappa }{2}^{-\tau -\kappa }{\pi }^{1∕2}\mathrm{\Gamma }(\tau )\mathrm{\Gamma }(1+\kappa +\ell )}{\begin{array}{c}\mathrm{\Gamma }(1-\kappa +\ell )\mathrm{\Gamma }(1∕2+\kappa ∕2-\ell ∕2+\tau ∕2)\hfill \\ \mathrm{\Gamma }(1∕2+\kappa ∕2+\ell ∕2+\tau ∕2)\hfill \end{array}}.$$

Then, the integral (3.30) becomes

$${(-1)}^{\kappa }{2}^{-\tau -\kappa }{\pi }^{1∕2}\frac{\mathrm{\Gamma }(1+\kappa +\ell )}{\mathrm{\Gamma }(1-\kappa +\ell )}\frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)\times \left[\frac{\mathrm{\Gamma }(\tau -\omega \zeta )}{\begin{array}{c}\mathrm{\Gamma }(1+\tau -\omega \zeta +\kappa -\ell )∕2)\hfill \\ \mathrm{\Gamma }(1+\tau -\omega \zeta +\kappa +\ell )∕2)\hfill \end{array}}\right]{(2^{-\kappa }Z)}^{-\zeta }d\zeta .$$

Now simplifying the equation, we get the desired result. □

Theorem 3.21.

provided $\tau >0$, $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$ and $\kappa \in \mathbb{N}\cup 0.$ where $A=(1-\tau ,\omega ),({𝔪}_1,{𝔄}_1,Y),{({𝔪}_j,{𝔄}_j)}_{2,N},{[{\rho }_j({𝔪}_{ji},{𝔄}_{ji})]}_{N+1,P_i}$ and $B={({𝔫}_j,{𝔅}_j)}_{1,M},(1∕2-\kappa ∕2+\ell ∕2-\tau ∕2,\omega ∕2),(-\kappa ∕2-\ell ∕2-\tau ∕2,\omega ∕2),{[{\rho }_j({𝔫}_{ji},{𝔅}_{ji})]}_{M+1,Q_i}.$

Proof. To prove it, do the same as in Theorem 5.1. □

3.4. Finite integrals associate the hypergeometric function and incomplete Aleph functions

The hypergeometric function ${}_2{F}_1$ is defined by

where ${(\alpha )}_n,{(\beta )}_n$ and ${(\gamma )}_n$ are Pochhammer symbols given as

$${(\mu )}_{\lambda }=\frac{\mathrm{\Gamma }(\mu +\lambda )}{\mathrm{\Gamma }(\mu )}=\left\{\begin{array}{cc}1\hfill & \mathrm{\text{if}}\lambda =0;\mu \in ℂ\setminus \{0\},\hfill \\ (\mu )(\mu +1)\dots (\mu +n-1)\hfill & \mathrm{\text{if}}\lambda =n\in \mathbb{N};\mu \in ℂ.\hfill \end{array}\right.$$(3.33)

Theorem 3.22.

which provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$.

Proof. Now write left-hand side of (3.34) in the form of Mellin–Barnes integral and after simplification, we have

Now putting $\delta =t+1\Rightarrow d\delta =dt,$ we obtain

$$\sum _{k=0}^{\infty }\frac{{(-1)}^k{(\ell +\tau -\omega )}_k{(\mathrm{\Lambda }+\tau -\omega )}_k}{k!{(\tau )}_k}\frac{1}{2\pi i}\times {\int }_L\frac{\mathrm{\Gamma }(\tau +k)\mathrm{\Gamma }(\omega +\zeta -\tau -k)}{\mathrm{\Gamma }(\omega +\zeta )}\phi (\zeta ,Y)Z^{-\zeta }d\zeta ,$$

after a small simplification, we get the desired result. □

Theorem 3.23.

which provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$.

Proof. To prove this theorem, do this as in Theorem 6.2. □

3.5. Finite integrals associate the Bessel Maitland function andincomplete Aleph functions

The Bessel Maitland function is a generalization of the Bessel function. It is also known as Wright generalized Bessel function and is defined by

Now, we establish the following integral that associates the incomplete Aleph functions with the Bessel Maitland function.

Theorem 3.24.

which provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$, $\tau -\kappa \tau >0$, $\tau >0$, $0<\kappa <1$ and $ℝ(\omega +1)>0$.

Proof. Here, we can write the left-hand side of (3.37) in the form of Mellin–Barnes integral and after simplification, we get

Use a formula given in Ref. 39 defined as

$${\int }_0^{\infty }{x}^t{J}_{\beta }^{\alpha }(x)dx=\frac{\mathrm{\Gamma }(t+1)}{\mathrm{\Gamma }(1+\beta -\alpha -\alpha t)}(ℝ(t)>-1,0<\alpha <1),$$

then we arrive at

$$\frac{1}{2\pi i}{\int }_L\phi (\zeta ,Y)\frac{\mathrm{\Gamma }(1+\omega -\tau \zeta )}{\mathrm{\Gamma }(1+\ell -\kappa -\kappa \omega +\kappa \tau \zeta )}{Z}^{-\zeta }d\zeta ,$$

after a small simplification, we get the desired result. □

Theorem 3.25.

which provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$, $\tau -\kappa \tau >0$, $\tau >0$, $0<\kappa <1$ and $ℝ(\omega +1)>0$.

Proof. To prove this, do the same as in Theorem 7.1. □

3.6. Finite integrals associate general class of polynomials and incomplete Aleph functions

Srivastava (see Eq. (7) on p. 185 of Ref. 40) introduced the general class of polynomials $S_{b_1,b_2,\dots ,b_n}^{a_1,a_2,\dots ,a_n}(x)$ and defined and represented it as follows :

where $b_1,\dots ,b_n$ are non-negative integers, $a_1,\dots ,a_n$ are arbitrary positive integers, and the coefficients $A_{b_i,k_i}$ (where $b_i$ and $k_i$ are both greater than or equal to 0) can be real or complex. By appropriately configuring the coefficients represented as $A_{b_i,k_i}$, the function $S_{b_1,b_2,\dots ,b_n}^{a_1,a_2,\dots ,a_n}(x)$ can encompass various well-established polynomials as specific instances. These notable instances encompass Laguerre Polynomials, Hermite Polynomials, Bessel Polynomials, Jacobi Polynomials, and other well-known cases detailed in Ref. 41, pp. 158–161. Now, we define the following integral that connects the incomplete Aleph function to this wide range of polynomials.

Theorem 3.26.

which converges under the conditions $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$, $\omega \ge 1$, $\tau \ge 1$, $\kappa \ge 0$, $\ell \ge 0$, $\beta \ge 0$, $p\ge 0$ (here, h and p are not simultaneous zeros) and $ℝ(\omega )+\beta min[ℝ(\frac{{𝔫}_j}{{𝔅}_j})]>0$ and $ℝ(\tau )+pmin[ℝ(\frac{{𝔫}_j}{{𝔅}_j})]>0$.

Proof. Utilizing Eqs. (2.5) and (8.1), and exchanging the sequence of summations and integration, we attain the anticipated outcome. □

4. Special Cases

In this section, our objective is to delve into a thorough evaluation of specific scenarios and circumstances that have emerged as a consequence of the results and Ref. 22. We have derived:

Case 1. If we replace a by $\eta -1$ and put $Y=\kappa =\ell =\omega =\tau =0,$ then formula (3.16) transforms into the following integral that incorporates the product of the Legendre function and the Aleph function:

provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}.$

Case 2. If we put $\omega$ by $\omega -1$ and $\tau$ by $\tau -1$, and put $Y=\kappa =\ell =0$, then formula (3.20) transforms into the following integral that incorporates the product of the Legendre function and the Aleph function:

provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}.$

Case 3. By replacing $\omega$ by $\omega -1$ and $\tau$ by $\tau -1$, and put $Y=c=d=0$, then integral (3.27) takes the following form:

provided $|\mathrm{\text{arg}}Z|<\frac{\pi \mathrm{\Omega }}{2}$, $ℝ[\omega +\mu min(\frac{{𝔢}_{𝔧}}{d_j})]>-1$ and $ℝ[\tau +\beta min(\frac{{𝔢}_{𝔧}}{d_j})]>-1(j=\overline{1,M}).$

5. Conclusion

The results derived in this context have fundamental significance and are expected to have practical uses in the analysis of both single-and multi-variable hyper-geometric series. These series have utility in diverse fields such as statistical mechanics, electrical networks, and probability theory. Moreover, all of the aforementioned results can be expressed in terms of incomplete H-functions, incomplete I-functions, I-function, and H-function.

Availability of Statistics and Materials

The availability of materials is already cited in the paper.

Funding Information

No funding available.

Statement of Disagreement

According to researchers, there are no conflicts of interest to disclose about the paper that is being presented.

Author’s Contribution

The study was directed by Ravi Shanker Dubey, who also analyzed the findings. Sachin Kumar organized the necessary research materials while Manvendra Narayan Mishra and Rahul Sharma prepared the paper and performed all the mathematical computations. The draft was read, corrected, and polished by all authors.