I. INTRODUCTION

Several integral formulas containing generalized special functions have been explored by many authors [15]. These integrals play a focal role in solving scientific and engineering problems. Here we develop Oberhettinger's integral formulas containing the product of general polynomials and incomplete H- functions. Many authors established several unified integral formulas containing a various kind of special functions [6-8]. The findings of this work are general in nature and very useful in science, engineering and finance. Here we find some special cases by specializing the parameters of general polynomials and incomplete H-functions (for example, Fox's H-function, Incomplete Fox-Wright functions, Fox-Wright functions and incomplete generalized hypergeometric functions) and also listed few known results. The main results of this work are very useful in solving the problems arising in digital signals, image processing, finance and ship target recognition by sonar system and radar signals [9-12].

The incomplete Gamma functions $\mathrm{\Gamma }(\xi ,z)$ and $\mathrm{{\rm Y}}$ $(\xi ,z)$ are defined as following:

And

Respectively, satisfy the decomposition formula given by:

The condition which we have used on the parameter $z$ in and anywhere else in this paper is unrestricted of $R(x)$ ( $x\in C$ ).

The incomplete generalized hypergeometric functions ${}_e{\mathrm{\Gamma }}_f$ and ${}_e{\gamma }_f$ are defined by Srivastava et al. [13] in terms of incomplete Gamma functions $\mathrm{\Gamma }(\mathbf{s},\mathbf{z})$ and $\mathrm{{\rm Y}}(s,z)$ as following:

And

• Where $\mathcal{L}$ is the Mellin-Barnes type contour having $\tau -i\mathrm{\infty }$ as the starting point and $\tau +i\mathrm{\infty }$ ( $\tau \in \mathbb{R}$ ) as
• the end point with the usual indentations to separate a set of poles from another of the integrand
• in each and every case.

The incomplete H- function ${\gamma }_{e,f}^{c,d}(x)$ and ${\mathrm{\Gamma }}_{e,f}^{c,d}(x)$ are introduced by Srivastava et. Al. [14] as following:

Where $f(s,z)=\frac{\mathrm{\Gamma }(1-{\mathbf{u}}_1-U_{1}s,z)\prod _{j=1}^c\mathrm{\Gamma }({\mathbf{v}}_j+V_{j}s)\prod _{j=2}^d\mathrm{\Gamma }(1-{\mathbf{u}}_j-U_{j}s)}{\prod _{j=c+1}^f\mathrm{\Gamma }(1-{\mathbf{v}}_j-V_{j}s)\prod _{j=d+1}^e\mathrm{\Gamma }({\mathbf{u}}_j+U_{j}s)}$

And

Where

The incomplete H- functions ${\gamma }_{e,f}^{c,d}(x)$ and ${\mathrm{\Gamma }}_{e,f}^{c,d}(x)$ in (6) and (7) respectively exist for all $z\ge 0$ under the same set of conditions and same set of contour stated in the articles presented by Kilbas et al. [15], Mathai and Saxena [16] and Mathai et al. [17].

Some special cases of incomplete H-function are as following:

(i)

If we take $z=0$ in (6), then the incomplete H-function ${\mathrm{\Gamma }}_{e,f}^{c,d}(x)$ reduces to Fox's H-function [18].

(ii) If we take $c=1$, $d=e$ and replace $f$ by $f+1$ and take suitable parameter, then the function (6) and (7) reduces to incomplete Fox-Wright function ${}_e{\mathrm{\Psi }}_f^{(\mathrm{\Gamma })}$ and ${}_e{\mathrm{\Psi }}_f^{(\gamma )}$ (for details see [14]).

And

(iii)

If we take $z=0$ in (9), then incomplete Fox-Wright function ${}_e{\mathrm{\Psi }}_f^{(\mathrm{\Gamma })}$ reduces to well-known Fox-Wright function ${}_e{\mathrm{\Psi }}_f$ (for details see [18]).

(iv) If we take ${\mathrm{U}}_{\mathrm{j}}={\mathrm{V}}_{\mathrm{k}}=1$ $(\mathrm{j}=1,\dots ,\mathrm{e},\mathrm{k}=1,\dots ,\mathrm{f})$ in (9) and (10), then incomplete Fox-Wright function reduces to the incomplete generalized hypergeometric functions ${}_e{\gamma }_f$ and ${}_e{\mathrm{\Gamma }}_f$ (see [13]).

And

The general polynomials are defined by Srivastava [19] as following:

Where ${\mathrm{n}}_{\mathrm{i}}=0,1,2\dots \mathrm{\forall }\mathrm{i}=(1,\dots ,\mathrm{s})$; $m_1,\dots ,m_s$ are arbitrary positive integers and the coefficients $A[{\mathrm{n}}_1,{\alpha }_1,\dots ;{\mathrm{n}}_s,{\alpha }_s]$ are arbitrary constants, real or complex.