Skip to main content

Recursion formulas for certain quadruple hypergeometric functions

Abstract

A remarkably large number of hypergeometric (and generalized) functions and a variety of their extensions have been presented and investigated in the literature by many authors. In this paper, we introduce five new hypergeometric functions in four variables and then establish several recursion formulas for these new functions. Some interesting particular cases and consequences of the main results are also considered.

Introduction and preliminaries

The ordinary hypergeometric functions have been the subject of extensive researches by several prominent mathematicians. These functions play a crucial role in mathematical analysis, physics, engineering and applied sciences. Most of the special functions, which have various physical and technical applications and are closely connected with orthogonal polynomial and problems of mechanical quadrature, can be expressed in terms of generalized hypergeometric functions. Agarwal et al. [1, 2] established some properties for the generalized Gauss hypergeometric functions, which were introduced by Özergin et al. Rahman et al. [3] defined further extensions of hypergeometric and Appell’s hypergeometric functions. Very recently, Saboor et al. [4] defined a new extension of Srivastava’s triple hypergeometric functions, and the authors presented some of their properties such as integral representations, derivative formulas, and recurrence relations.

Many modern mathematics and theoretical physics problems lead to the study of the hypergeometric functions of several complex variables (see, e.g., [516]). These include, for example, problems in the representation theory, combinatorics, number theory, analytic continuation of integrals of the Mellin–Barnes type, and algebraic geometry. Moreover, hypergeometric-type functions are seen in several applications of physical and chemical problems [1720].

In [21], Exton defined 21 complete hypergeometric functions in four variables denoted by the symbols \(K_{1},K_{2},\dots ,K_{21}\). In [22], Sharma and Parihar introduced 83 complete quadruple hypergeometric functions, denoted by \(F_{1}^{(4)},F^{(4)}_{2},\dots ,F^{(4)}_{83}\). Bin-Saad and Younis [23, 24] introduced 30 new quadruple hypergeometric functions written as \(X_{1}^{(4)}, X_{2}^{(4)},\dots ,X_{30}^{(4)}\). In [25] Younis et al. discovered the existence of six additional complete hypergeometric functions in four variables, \(X_{85}^{(4)}, X_{86}^{(4)},\dots ,X_{90}^{(4)}\). Motivated by the above investigations, we introduce five new quadruple hypergeometric functions as follows:

$$\begin{aligned}& X_{95}^{(4)}(\rho _{1}, \rho _{1}, \rho _{3}, \rho _{5}, \rho _{1}, \rho _{2}, \rho _{4}, \rho _{6}; \omega _{2}, \omega _{1} , \omega _{1}, \omega _{1} ; x, y, z, u) \\& \quad =\sum_{m, n,p,q=0}^{\infty } \frac{(\rho _{1})_{2m+n}(\rho _{2})_{n}(\rho _{3})_{p}(\rho _{4})_{p}(\rho _{5})_{q}(\rho _{6})_{q}}{(\omega _{1})_{n+p+q}(\omega _{2})_{m}} \frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!} \frac{u^{q}}{q!} \\& \qquad \biggl( \vert x \vert < \frac{1}{4}, \vert y \vert < 1, \vert z \vert < 1, \vert t \vert < 1 \biggr), \end{aligned}$$
(1.1)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{1}, \rho _{1}, \rho _{2}, \rho _{3}, \rho _{1}, \rho _{2}, \rho _{2}, \rho _{4}; \omega _{1}, \omega _{2} , \omega _{1}, \omega _{2} ; x, y, z, u) \\& \quad =\sum_{m, n,p,q=0}^{\infty } \frac{(\rho _{1})_{2m+n}(\rho _{2})_{2p+n}(\rho _{3})_{q}(\rho _{4})_{q}}{(\omega _{1})_{m+p}(\omega _{2})_{n+q}} \frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!} \frac{u^{q}}{q!} \\& \qquad \biggl( \vert x \vert < \frac{1}{4}, \vert y \vert < 1, \vert z \vert < \frac{1}{4}, \vert t \vert < 1 \biggr), \end{aligned}$$
(1.2)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{1}, \rho _{1}, \rho _{2}, \rho _{3}, \rho _{1}, \rho _{2}, \rho _{2}, \rho _{4}; \omega _{1}, \omega _{1} , \omega _{2}, \omega _{1} ; x, y, z, u) \\& \quad =\sum_{m, n,p,q=0}^{\infty } \frac{(\rho _{1})_{2m+n}(\rho _{2})_{2p+n}(\rho _{3})_{q}(\rho _{4})_{q}}{(\omega _{1})_{m+n+q}(\omega _{2})_{p}} \frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!} \frac{u^{q}}{q!} \\& \qquad \biggl( \vert x \vert < \frac{1}{4}, \vert y \vert < 1, \vert z \vert < \frac{1}{4}, \vert t \vert < 1 \biggr), \end{aligned}$$
(1.3)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{1}, \rho _{1}, \rho _{2}, \rho _{3}, \rho _{1}, \rho _{2}, \rho _{2}, \rho _{4}; \omega _{1} , \omega _{2}, \omega _{1}, \omega _{1} ; x, y, z, u) \\& \quad =\sum_{m, n,p,q=0}^{\infty } \frac{(\rho _{1})_{2m+n}(\rho _{2})_{2p+n}(\rho _{3})_{q}(\rho _{4})_{q}}{(\omega _{1})_{m+p+q}(\omega _{2})_{n}} \frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!} \frac{u^{q}}{q!} \\& \qquad \biggl( \vert x \vert < \frac{1}{4}, \vert y \vert < 1, \vert z \vert < \frac{1}{4}, \vert t \vert < 1 \biggr), \end{aligned}$$
(1.4)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{1}, \rho _{1}, \rho _{2}, \rho _{3}, \rho _{1}, \rho _{2}, \rho _{2}, \rho _{4}; c , c, c, c ; x, y, z, u) \\& \quad =\sum_{m, n,p,q=0}^{\infty } \frac{(\rho _{1})_{2m+n}(\rho _{2})_{2p+n}(\rho _{3})_{q}(\rho _{4})_{q}}{(c)_{m+n+p+q}} \frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!} \frac{u^{q}}{q!} \\& \qquad \biggl( \vert x \vert < \frac{1}{4}, \vert y \vert < 1, \vert z \vert < \frac{1}{4}, \vert t \vert < 1 \biggr), \end{aligned}$$
(1.5)

where \((a)_{m}\) is the Pochhammer symbol defined by

$$ (a)_{m}=\frac{\Gamma (a+m)}{\Gamma (a)}=a(a+1)\cdots (a+m-1), $$

for \(m\geq {1}\), \((a)_{0}=1\), and Γ being the well-known Gamma function.

Recently, many authors have obtained several recursion formulas involving classical hypergeometric functions. In [26], Opps et al. introduced the recursion formulas for Appell’s function \(F_{2}\) and gave its applications to radiation field problems. Wang [27] presented the recursion formulas for Appell functions \(F_{1}\), \(F_{2}\), \(F_{3}\), and \(F_{4}\). Sahai and Verma [28, 29] established the recursion formulas for Lauricella’s triple functions, Srivastava hypergeometric functions in three variables, k-variable Lauricella functions, and the Srivastava–Daoust and related multivariable hypergeometric functions. In [25, 30], the authors gave the recursion formulas for Srivastava general triple hypergeometric function and Exton’s triple hypergeometric functions. In this present paper, we establish several recursion formulas associated with the quadruple functions (1.1)–(1.5).

In the following, some abbreviated notations are used in this paper. We, for example, write \(X_{95}^{(4)}\) for the series

$$ X_{95}^{(4)}(\rho _{1}, \rho _{1}, \rho _{3}, \rho _{5}, \rho _{1}, \rho _{2}, \rho _{4}, \rho _{6}; \omega _{2}, \omega _{1} , \omega _{1}, \omega _{1} ; x, y, z, u) $$

and \(X_{95}^{(4)}(\rho _{1}+n)\) for

$$ X_{95}^{(4)}(\rho _{1}+n, \rho _{1}+n, \rho _{3}, \rho _{5}, \rho _{1}+n, \rho _{2}, \rho _{4}, \rho _{6}; \omega _{2}, \omega _{1} , \omega _{1}, \omega _{1} ; x, y, z, u). $$

The notation \(X_{95}^{(4)}(\rho _{1}+n, \rho _{2}+n_{1})\) stands for

$$ X_{95}^{(4)}(\rho _{1}+n, \rho _{1}+n, \rho _{3}, \rho _{5}, \rho _{1}+n, \rho _{2}+n_{1}, \rho _{4}, \rho _{6}; \omega _{2}, \omega _{1} , \omega _{1}, \omega _{1} ; x, y, z, u) $$

and \(X_{95}^{(4)}(\rho _{1}+n, \rho _{2}+n_{1}, \omega _{1}+n_{2})\) stands for

$$ X_{95}^{(4)}(\rho _{1}+n, \rho _{1}+n, \rho _{3}, \rho _{5}, \rho _{1}+n, \rho _{2}+n_{1}, \rho _{4}, \rho _{6}; \omega _{2}, \omega _{1}+n_{2} , \omega _{1}+n_{2}, \omega _{1}+n_{2} ; x, y, z, u), $$

etc.

Main results

Here, we present certain recursion formulas for the hypergeometric functions of four variables \(X_{95}^{(4)}, X_{96}^{(4)},\dots ,X_{99}^{(4)}\). Throughout the paper, n denotes a nonnegative integer.

Theorem 2.1

The following recursion formulas hold true for the numerator parameters \(\rho _{1}\), \(\rho _{2}\), \(\rho _{3}\), \(\rho _{4}\), \(\rho _{5}\), \(\rho _{6}\) of \(X_{95}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] X_{95}^{(4)}(\rho _{1}+n)={}&X_{95}^{(4)}+ \frac{2x}{\omega _{2}}\sum_{n_{1}=1}^{n}( \rho _{1}+n_{1})X_{95}^{(4)}(\rho _{1}+1+n_{1}, \omega _{2}+1) \\ &{}+\frac{y\rho _{2}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+1, \omega _{1}+1), \end{aligned} \end{aligned}$$
(2.1)
$$\begin{aligned}& \begin{aligned}[b] X_{95}^{(4)}(\rho _{1}-n)={}&X_{95}^{(4)}- \frac{2x}{\omega _{2}}\sum_{n_{1}=1}^{n}( \rho _{1}+1-n_{1})X_{95}^{(4)}( \rho _{1}+2-n_{1}, \omega _{2}+1) \\ &{}-\frac{y\rho _{2}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{1}+1-n_{1}, \rho _{2}+1, \omega _{1}+1), \end{aligned} \end{aligned}$$
(2.2)
$$\begin{aligned}& \begin{aligned}[b] X_{95}^{(4)}(\rho _{2}+n)={}&X_{95}^{(4)}+ \frac{y\rho _{1}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{1}+ 1, \rho _{2}+n_{1}, c+1) \\ &{}+\frac{z\rho _{3}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{2}+n_{1}, \rho _{3}+1, c+1), \end{aligned} \end{aligned}$$
(2.3)
$$\begin{aligned}& \begin{aligned}[b] X_{95}^{(4)}(\rho _{2}-n)={}&X_{95}^{(4)}- \frac{y\rho _{1}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{1}+1, \rho _{2}+1-n_{1}, c+1) \\ &{}-\frac{z\rho _{3}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{2}+1-n_{1}, \rho _{3}+1, c+1), \end{aligned} \end{aligned}$$
(2.4)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{3}+n)=X_{95}^{(4)}+ \frac{z\rho _{2}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{2}+1, \rho _{3}+n_{1}, c+1), \end{aligned}$$
(2.5)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{3}-n)=X_{95}^{(4)}- \frac{z\rho _{2}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{2}+1, \rho _{3}+1-n_{1}, c+1), \end{aligned}$$
(2.6)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{4}+n)=X_{95}^{(4)}+ \frac{u\rho _{5}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{5}+1, \rho _{4}+n_{1}, c+1), \end{aligned}$$
(2.7)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{4}-n)=X_{95}^{(4)}- \frac{u\rho _{5}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{5}+1, \rho _{4}+1-n_{1}, c+1), \end{aligned}$$
(2.8)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{5}+n)=X_{95}^{(4)}+ \frac{u\rho _{4}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{4}+1, \rho _{5}+n_{1}, c+1), \end{aligned}$$
(2.9)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{5}-n)=X_{95}^{(4)}- \frac{u\rho _{4}}{c}\sum_{n_{1}=1}^{n}X_{95}^{(4)}( \rho _{4}+1, \rho _{5}+1-n_{1}, c+1). \end{aligned}$$
(2.10)

Proof

From the definition of the hypergeometric function \(X_{95}^{(4)}\) and the relation

$$ (\rho _{1}+1)_{2m+n}=(\rho _{1})_{2m+n} \biggl(1+\frac{2m}{\rho _{1}}+ \frac{n}{\rho _{1}} \biggr), $$
(2.11)

we obtain the following contiguous relation:

$$ \begin{aligned}[b] X_{95}^{(4)}(\rho _{1}+1)={}& X_{95}^{(4)} + \frac{2x}{\omega _{2}}( \rho _{1}+1) X_{95}^{(4)}(\rho _{1}+2, \omega _{2}+1) \\ &{}+\frac{y\rho _{2}}{\omega _{1}}X_{95}^{(4)}(\rho _{1}+1, \rho _{2}+1, \omega _{1}+1). \end{aligned} $$
(2.12)

To find a contiguous relation for \(X_{95}^{(4)}(\rho _{1}+2)\), we replace \(\rho _{1}\rightarrow \rho _{1}+1\) in (2.12) and simplify. This leads to

$$ \begin{aligned}[b] X_{95}^{(4)}(\rho _{1}+2)={}& X_{95}^{(4)}+ \frac{2x}{\omega _{2}} \sum_{n_{1}=1}^{2}(\rho _{1}+n_{1}) X_{95}^{(4)}( \rho _{1}+n_{1}+1, \omega _{2}+1) \\ &{}+\frac{y\rho _{2}}{\omega _{1}}\sum_{n_{1}=1}^{2}X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+1, \omega _{1}+1). \end{aligned} $$
(2.13)

Iterating this process n times, we obtain (2.1). For the proof of (2.2), replace the parameter \(\rho _{1}\rightarrow \rho _{1}-1\) in (2.12). This gives

$$ X_{95}^{(4)}(\rho _{1}-1) = X_{1} - \frac{2x}{\omega _{2}}\rho _{1} X_{95}^{(4)}(\rho _{1}+1 ,\omega _{2}+1)- \frac{y\rho _{2}}{\omega _{1}} X_{95}^{(4)}( \rho _{2}+1, \omega _{1}+1). $$
(2.14)

Iteratively, we get (2.2).

The recursion formulas (2.3)–(2.10) can be proved in a similar manner. □

Theorem 2.2

The following recursion formulas hold true for the numerator parameters \(\rho _{2}\), \(\rho _{3}\), \(\rho _{4}\), \(\rho _{5}\) of \(X_{95}^{(4)}\):

$$\begin{aligned}& X_{95}^{(4)}(\rho _{2}+n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{1})_{n_{1}} y^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.15)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{2}-n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{1})_{n_{1}} (-y)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{1}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.16)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{3}+n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} z^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.17)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{3}-n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} (-z)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.18)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{4}+n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} z^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.19)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{4}-n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} (-z)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{3}+n_{1}, c+n_{1}), \end{aligned}$$
(2.20)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{5}+n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{6})_{n_{1}} u^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{5}+n_{1}, \rho _{6}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.21)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{5}-n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{6})_{n_{1}} (-u)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{6}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.22)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{6}+n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{5})_{n_{1}} u^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{5}+n_{1}, \rho _{6}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.23)
$$\begin{aligned}& X_{95}^{(4)}(\rho _{6}-n) = \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{5})_{n_{1}} (-u)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{5}+n_{1}, \omega _{1}+n_{1}). \end{aligned}$$
(2.24)

Proof

The proof of (2.15) is based upon the principle of mathematical induction on \(n\in \mathbb{N}\). For \(n=1\), the result (2.15) is true obviously by (2.3). Suppose (2.15) is true for \(n=m\), that is,

$$ X_{95}^{(4)}(\rho _{2}+m) = \sum _{n_{1}\leq m}\binom{n}{n_{1}} \frac{(\rho _{1})_{n_{1}} y^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+n_{1}, \omega _{1}+n_{1}). $$
(2.25)

Replacing \(\rho _{2}\mapsto \rho _{2}+1\) in (2.25) and using the contiguous relation (2.3) for \(n=1\), we get

$$ \begin{aligned}[b] &X_{95}^{(4)}(\rho _{2}+m+1) \\ &\quad =\sum_{n_{1}\leq m}\binom{n}{n_{1}} \frac{(\rho _{1})_{n_{1}} y^{n_{1}}}{(\omega _{1})_{n_{1}}} \biggl\{ X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+n_{1}, \omega _{1}+n_{1}) \\ &\qquad {} +\frac{(\rho _{1}+n_{1})y}{(\omega _{1}+n_{1})}X_{95}^{(4)} ( \rho _{1}+n_{1}+1, \rho _{2}+n_{1}, \omega _{1}+n_{1}+1 ) \biggr\} . \end{aligned} $$
(2.26)

Simplifying, (2.26) takes the form

$$ \begin{aligned}[b] &X_{95}^{(4)} ({a}_{2}+m+1 ) \\ &\quad =\sum_{n_{1}\leq m}\binom{n}{n_{1}} \frac{(\rho _{1})_{n_{1}}y^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+n_{1}, \omega _{1}+n_{1}) \\ &\qquad {} +\sum_{n_{1}\leq m+1}\binom{n}{n_{1}-1} \frac{(\rho _{1})_{n_{1}} y^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+n_{1}, \omega _{1}+n_{1}). \end{aligned} $$
(2.27)

Using Pascal’s identity in (2.27), we have

$$ X_{95}^{(4)}(\rho _{2}+m+1)= \sum _{n_{1}\leq m+1}\binom{n}{n_{1}} \frac{(\rho _{1})_{n_{1}} y^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{95}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+n_{1}, \omega _{1}+n_{1}). $$
(2.28)

This establishes (2.15) for \(n=m+1\). Hence, by induction, the result given in (2.15) is true for all values of n. The recursion formulas (2.16)–(2.24) can be proved in a similar manner. □

Theorem 2.3

The following recursion formulas hold true for the denominator parameter c of \(X_{95}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] &X_{95}^{(4)}(\omega _{1}-n) \\ &\quad = X_{95}^{(4)} + \rho _{1} \rho _{2} y\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{95}^{(4)}( \rho _{1}+1, \rho _{2}+1, \omega _{1}+2-n_{1}) \\ &\qquad {} +{\rho _{3} \rho _{4} z}\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{95}^{(4)}( \rho _{3}+1, \rho _{4}+1, \omega _{1}+2-n_{1}) \\ &\qquad {} +{\rho _{5} \rho _{6} u}\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{95}^{(4)}( \rho _{5}+1, \rho _{6}+1, \omega _{1}+2-n_{1}), \end{aligned} \end{aligned}$$
(2.29)
$$\begin{aligned}& \begin{aligned}[b] &X_{95}^{(4)}(\omega _{2}-n) \\ &\quad = X_{95}^{(4)} +{(\rho _{1})_{2} x}\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{2} - n_{1})(\omega _{2}+1-n_{1})} X_{95}^{(4)}( \rho _{1}+2, \omega _{2}+2-n_{1}). \end{aligned} \end{aligned}$$
(2.30)

Proof

Using the definition of hypergeometric function \(X_{95}^{(4)}\) and the relation

$$ \frac{1}{(\omega _{1}-1)_{n+p+q}}= \frac{1}{(\omega _{1})_{n+p+q}} \biggl(1+\frac{n}{\omega _{1}-1}+ \frac{p}{\omega _{1}-1}+ \frac{q}{\omega _{1}-1} \biggr), $$
(2.31)

we have

$$ \begin{aligned}[b] X_{95}^{(4)}(\omega _{1}-1)={}&X_{95}^{(4)} + \frac{\rho _{1} \rho _{2} y}{\omega _{1}(\omega _{1}-1)}X_{95}^{(4)}( \rho _{1}+1, \rho _{2}+1, \omega _{1}+1) \\ &{}+\frac{\rho _{3} \rho _{4} z}{\omega _{1}(\omega _{1}-1)}X_{95}^{(4)}( \rho _{3}+1, \rho _{4}+1, \omega _{1}+2-n_{1}) \\ &{}+ \frac{\rho _{5} \rho _{6} u}{\omega _{1}(\omega _{1}-1)}X_{95}^{(4)}( \rho _{5}+1, \rho _{6}+1, \omega _{1}+2-n_{1}). \end{aligned} $$
(2.32)

Using this contiguous relation for \(X_{95}^{(4)}\) with the parameter \(\omega _{1}-n\) for n times, we get (2.29). Recursion formula (2.30) can be proved in a similar manner. □

Theorem 2.4

The following recursion formulas hold true for the numerator parameters \(\rho _{1}\), \(\rho _{2}\), \(\rho _{3}\), \(\rho _{4}\) of \(X_{96}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] X_{96}^{(4)}(\rho _{1}+n)={}&X_{96}^{(4)}+ \frac{2x}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{1}+n_{1})X_{96}^{(4)}(\rho _{1}+1+n_{1}, \omega _{1}+1) \\ &{}+\frac{y\rho _{2}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+1, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.33)
$$\begin{aligned}& \begin{aligned}[b] X_{96}^{(4)}(\rho _{1}-n)={}&X_{96}^{(4)}- \frac{2x}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{1}+1-n_{1})X_{96}^{(4)}( \rho _{1}+2-n_{1}, \omega _{1}+1) \\ &{}-\frac{y\rho _{2}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{1}+1-n_{1}, \rho _{2}+1, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.34)
$$\begin{aligned}& \begin{aligned}[b] X_{96}^{(4)}(\rho _{2}+n)={}&X_{96}^{(4)}+ \frac{2z}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{2}+n_{1})X_{96}^{(4)}(\rho _{2}+ 1+n_{1}, \omega _{1}+1) \\ &{}+\frac{y\rho _{1}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{1}+1, \rho _{2}+n_{1}, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.35)
$$\begin{aligned}& \begin{aligned} X_{96}^{(4)}(\rho _{2}-n)={}&X_{96}^{(4)}- \frac{2z}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{2}+1-n_{1})X_{96}^{(4)}(\rho _{2}+2-n_{1}, \omega _{1}+1) \\ &{}-\frac{y\rho _{1}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{1}+1, \rho _{2}+1-n_{1}, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.36)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{3}+n)=X_{96}^{(4)}+ \frac{z\rho _{4}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{4}+1, \rho _{3}+n_{1}, \omega _{2}+1), \end{aligned}$$
(2.37)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{3}-n)=X_{96}^{(4)}- \frac{z\rho _{4}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{4}+1, \rho _{3}+1-n_{1}, \omega _{2}+1), \end{aligned}$$
(2.38)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{4}+n)=X_{96}^{(4)}+ \frac{u\rho _{3}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{3}+1, \rho _{4}+n_{1}, \omega _{2}+1), \end{aligned}$$
(2.39)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{4}-n)=X_{96}^{(4)}- \frac{u\rho _{3}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{96}^{(4)}( \rho _{3}+1, \rho _{4}+1-n_{1}, \omega _{2}+1). \end{aligned}$$
(2.40)

Theorem 2.5

The following recursion formulas hold true for the numerator parameters \(\rho _{3}\), \(\rho _{4}\) of \(X_{96}^{(4)}\):

$$\begin{aligned}& X_{96}^{(4)}(\rho _{3}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} z^{n_{1}} }{(\omega _{2})_{n_{1}}} X_{96}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{2}+n_{1}), \end{aligned}$$
(2.41)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{3}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} (-z)^{n_{1}}}{(\omega _{2})_{n_{1}}} X_{96}^{(4)}( \rho _{4}+n_{1}, \omega _{2}+n_{1}), \end{aligned}$$
(2.42)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{4}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} u^{n_{1}} }{(\omega _{2})_{n_{1}}} X_{96}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{2}+n_{1}), \end{aligned}$$
(2.43)
$$\begin{aligned}& X_{96}^{(4)}(\rho _{4}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} (-u)^{n_{1}}}{(\omega _{2})_{n_{1}}} X_{96}^{(4)}( \rho _{3}+n_{1}, \omega _{2}+n_{1}). \end{aligned}$$
(2.44)

Theorem 2.6

The following recursion formulas hold true for the denominator parameters \(\omega _{1}\), \(\omega _{2}\) of \(X_{96}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] &X_{96}^{(4)}(\omega _{1}-n) \\ &\quad = X_{96}^{(4)} + (\rho _{1})_{2} x\sum_{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{96}^{(4)}( \rho _{1}+2, \omega _{1}+2-n_{1}) \\ &\qquad {} +{(\rho _{2})_{2} z}\sum_{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{96}^{(4)}( \rho _{2}+2, \omega _{1}+2-n_{1}), \end{aligned} \end{aligned}$$
(2.45)
$$\begin{aligned}& \begin{aligned}[b] &X_{96}^{(4)}(\omega _{2}-n) \\ &\quad = X_{96}^{(4)} + \rho _{1}\rho _{2} y\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{2} - n_{1})(\omega _{2}+1-n_{1})} X_{96}^{(4)}( \rho _{1}+1, \rho _{2}+1, \omega _{2}+2-n_{1}) \\ &\qquad {} + \rho _{3} \rho _{4} u\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{2} - n_{1})(\omega _{2}+1-n_{1})} X_{96}^{(4)}( \rho _{3}+1, \rho _{4}+1, \omega _{2}+2-n_{1}). \end{aligned} \end{aligned}$$
(2.46)

Theorem 2.7

The following recursion formulas hold true for the numerator parameters \(\rho _{1}\), \(\rho _{2}\), \(\rho _{3}\), \(\rho _{4}\) of \(X_{97}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] X_{97}^{(4)}(\rho _{1}+n)=&{}X_{97}^{(4)}+ \frac{2x}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{1}+n_{1})X_{97}^{(4)}(\rho _{1}+1+n_{1}, \omega _{1}+1) \\ &{}+\frac{y\rho _{2}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+1, \omega _{1}+1), \end{aligned} \end{aligned}$$
(2.47)
$$\begin{aligned}& \begin{aligned}[b] X_{97}^{(4)}(\rho _{1}-n)={}&X_{97}^{(4)}- \frac{2x}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{1}+1-n_{1})X_{97}^{(4)}( \rho _{1}+2-n_{1}, \omega _{1}+1) \\ &{}-\frac{y\rho _{2}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{1}+1-n_{1}, \rho _{2}+1, \omega _{1}+1), \end{aligned} \end{aligned}$$
(2.48)
$$\begin{aligned}& \begin{aligned}[b] X_{97}^{(4)}(\rho _{2}+n)={}&X_{97}^{(4)}+ \frac{2z}{\omega _{2}}\sum_{n_{1}=1}^{n}( \rho _{2}+n_{1})X_{97}^{(4)}(\rho _{2}+ 1+n_{1}, \omega _{2}+1) \\ &{}+\frac{y\rho _{1}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{1}+1, \rho _{2}+n_{1}, \omega _{1}+1), \end{aligned} \end{aligned}$$
(2.49)
$$\begin{aligned}& \begin{aligned}[b] X_{97}^{(4)}(\rho _{2}-n)={}&X_{97}^{(4)}- \frac{2z}{\omega _{2}}\sum_{n_{1}=1}^{n}( \rho _{2}+1-n_{1})X_{97}^{(4)}(\rho _{2}+2-n_{1}, \omega _{2}+1) \\ &{}-\frac{y\rho _{1}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{1}+1, \rho _{2}+1-n_{1}, \omega _{1}+1), \end{aligned} \end{aligned}$$
(2.50)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{3}+n)=X_{97}^{(4)}+ \frac{u\rho _{4}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{4}+1, \rho _{3}+n_{1}, \omega _{1}+1), \end{aligned}$$
(2.51)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{3}-n)=X_{97}^{(4)}- \frac{u\rho _{4}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{4}+1, \rho _{3}+1-n_{1}, \omega _{1}+1), \end{aligned}$$
(2.52)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{4}+n)=X_{97}^{(4)}+ \frac{u\rho _{3}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{3}+1, \rho _{4}+n_{1}, \omega _{1}+1), \end{aligned}$$
(2.53)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{4}-n)=X_{97}^{(4)}- \frac{u\rho _{3}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{97}^{(4)}( \rho _{3}+1, \rho _{4}+1-n_{1}, \omega _{1}+1). \end{aligned}$$
(2.54)

Theorem 2.8

The following recursion formulas hold true for the numerator parameters \(\rho _{3}\), \(\rho _{4}\) of the \(X_{97}^{(4)}\):

$$\begin{aligned}& X_{97}^{(4)}(\rho _{3}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} u^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{97}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.55)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{3}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} (-u)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{97}^{(4)}( \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.56)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{4}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} u^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{97}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.57)
$$\begin{aligned}& X_{97}^{(4)}(\rho _{4}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} (-u)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{97}^{(4)}( \rho _{3}+n_{1}, \omega _{1}+n_{1}). \end{aligned}$$
(2.58)

Theorem 2.9

The following recursion formulas hold true for the denominator parameters \(\omega _{1}\), \(\omega _{2}\) of \(X_{97}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] &X_{97}^{(4)}(\omega _{1}-n) \\ &\quad = X_{97}^{(4)} + (\rho _{1})_{2} x\sum_{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{97}^{(4)}( \rho _{1}+2, \omega _{1}+2-n_{1}) \\ &\qquad {} + \rho _{1} \rho _{2} y\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{97}^{(4)}( \rho _{1}+1, \rho _{2}+1, \omega _{1}+2-n_{1}) \\ &\qquad {} + \rho _{3} \rho _{4} u\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{97}^{(4)}( \rho _{3}+1, \rho _{4}+1, \omega _{1}+2-n_{1}), \end{aligned} \end{aligned}$$
(2.59)
$$\begin{aligned}& \begin{aligned}[b] &X_{97}^{(4)}(\omega _{2}-n) \\ &\qquad = X_{97}^{(4)} + (\rho _{2})_{2} z\sum_{n_{1}=1}^{n} \frac{1}{(\omega _{2} - n_{1})(\omega _{2}+1-n_{1})} X_{97}^{(4)}( \rho _{2}+2, \omega _{2}+2-n_{1}). \end{aligned} \end{aligned}$$
(2.60)

Theorem 2.10

The following recursion formulas hold true for the numerator parameters \(\rho _{1}\), \(\rho _{2}\), \(\rho _{3}\), \(\rho _{4}\) of \(X_{98}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] X_{98}^{(4)}(\rho _{1}+n)={}&X_{98}^{(4)}+ \frac{2x}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{1}+n_{1})X_{98}^{(4)}(\rho _{1}+1+n_{1}, \omega _{1}+1) \\ &{}+\frac{y\rho _{2}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+1, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.61)
$$\begin{aligned}& \begin{aligned}[b] X_{98}^{(4)}(\rho _{1}-n)={}&X_{98}^{(4)}- \frac{2x}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{1}+1-n_{1})X_{98}^{(4)}( \rho _{1}+2-n_{1}, \omega _{1}+1) \\ &-\frac{y\rho _{2}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{1}+1-n_{1}, \rho _{2}+1, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.62)
$$\begin{aligned}& \begin{aligned}[b] X_{98}^{(4)}(\rho _{2}+n)={}&X_{98}^{(4)}+ \frac{2z}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{2}+n_{1})X_{98}^{(4)}(\rho _{2}+ 1+n_{1}, \omega _{1}+1) \\ &{}+\frac{y\rho _{1}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{1}+1, \rho _{2}+n_{1}, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.63)
$$\begin{aligned}& \begin{aligned}[b] X_{98}^{(4)}(\rho _{2}-n)={}&X_{98}^{(4)}- \frac{2z}{\omega _{1}}\sum_{n_{1}=1}^{n}( \rho _{2}+1-n_{1})X_{98}^{(4)}(\rho _{2}+2-n_{1}, \omega _{1}+1) \\ &{}-\frac{y\rho _{1}}{\omega _{2}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{1}+1, \rho _{2}+1-n_{1}, \omega _{2}+1), \end{aligned} \end{aligned}$$
(2.64)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{3}+n)=X_{98}^{(4)}+ \frac{u\rho _{4}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{4}+1, \rho _{3}+n_{1}, \omega _{1}+1), \end{aligned}$$
(2.65)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{3}-n)=X_{98}^{(4)}- \frac{u\rho _{4}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{4}+1, \rho _{3}+1-n_{1}, \omega _{1}+1), \end{aligned}$$
(2.66)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{4}+n)=X_{98}^{(4)}+ \frac{u\rho _{3}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{3}+1, \rho _{4}+n_{1}, \omega _{1}+1), \end{aligned}$$
(2.67)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{4}-n)=X_{98}^{(4)}- \frac{u\rho _{3}}{\omega _{1}}\sum_{n_{1}=1}^{n}X_{98}^{(4)}( \rho _{3}+1, \rho _{4}+1-n_{1}, \omega _{1}+1). \end{aligned}$$
(2.68)

Theorem 2.11

The following recursion formulas hold true for the numerator parameters \(\rho _{3}\), \(\rho _{4}\) of \(X_{98}^{(4)}\):

$$\begin{aligned}& X_{98}^{(4)}(\rho _{3}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} u^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{98}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.69)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{3}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} (-u)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{98}^{(4)}( \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.70)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{4}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} u^{n_{1}} }{(\omega _{1})_{n_{1}}} X_{98}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, \omega _{1}+n_{1}), \end{aligned}$$
(2.71)
$$\begin{aligned}& X_{98}^{(4)}(\rho _{4}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} (-u)^{n_{1}}}{(\omega _{1})_{n_{1}}} X_{98}^{(4)}( \rho _{3}+n_{1}, \omega _{1}+n_{1}). \end{aligned}$$
(2.72)

Theorem 2.12

The following recursion formulas hold true for the denominator parameters \(\omega _{1}\), \(\omega _{2}\) of \(X_{98}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] &X_{98}^{(4)}(\omega _{1}-n) \\ &\quad = X_{98}^{(4)} + (\rho _{1})_{2} x\sum_{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{98}^{(4)}( \rho _{1}+2, \omega _{1}+2-n_{1}) \\ &\qquad {} + (\rho _{2})_{2} z\sum_{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{98}^{(4)}( \rho _{2}+2, \omega _{1}+2-n_{1}) \\ &\qquad {} + \rho _{3} \rho _{4} u\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{1} - n_{1})(\omega _{1}+1-n_{1})} X_{98}^{(4)}( \rho _{3}+1, \rho _{4}+1, \omega _{1}+2-n_{1}), \end{aligned} \end{aligned}$$
(2.73)
$$\begin{aligned}& \begin{aligned}[b] &X_{98}^{(4)}(\omega _{2}-n) \\ &\quad = X_{98}^{(4)} + \rho _{1} \rho _{2} y\sum _{n_{1}=1}^{n} \frac{1}{(\omega _{2} - n_{1})(\omega _{2}+1-n_{1})} X_{98}^{(4)}( \rho _{1}+1, \rho _{2}+1, \omega _{2}+2-n_{1}). \end{aligned} \end{aligned}$$
(2.74)

Theorem 2.13

The following recursion formulas hold true for the numerator parameters \(\rho _{1}\), \(\rho _{2}\), \(\rho _{3}\), \(\rho _{4}\) of \(X_{99}^{(4)}\):

$$\begin{aligned}& \begin{aligned}[b] X_{99}^{(4)}(\rho _{1}+n)={}&X_{99}^{(4)}+ \frac{2x}{c}\sum_{n_{1}=1}^{n}( \rho _{1}+n_{1})X_{99}^{(4)}(\rho _{1}+1+n_{1}, c+1) \\ &{}+\frac{y\rho _{2}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{1}+n_{1}, \rho _{2}+1, c+1), \end{aligned} \end{aligned}$$
(2.75)
$$\begin{aligned}& \begin{aligned}[b] X_{99}^{(4)}(\rho _{1}-n)={}&X_{99}^{(4)}- \frac{2x}{c}\sum_{n_{1}=1}^{n}( \rho _{1}+1-n_{1})X_{99}^{(4)}( \rho _{1}+2-n_{1}, c+1) \\ &{}-\frac{y\rho _{2}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{1}+1-n_{1}, \rho _{2}+1, c+1), \end{aligned} \end{aligned}$$
(2.76)
$$\begin{aligned}& \begin{aligned}[b] X_{99}^{(4)}(\rho _{2}+n)={}&X_{99}^{(4)}+ \frac{2z}{c}\sum_{n_{1}=1}^{n}( \rho _{2}+n_{1})X_{99}^{(4)}(\rho _{2}+ 1+n_{1}, c+1) \\ &{}+\frac{y\rho _{1}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{1}+1, \rho _{2}+n_{1}, c+1), \end{aligned} \end{aligned}$$
(2.77)
$$\begin{aligned}& \begin{aligned}[b] X_{99}^{(4)}(\rho _{2}-n)={}&X_{99}^{(4)}- \frac{2z}{c}\sum_{n_{1}=1}^{n}( \rho _{2}+1-n_{1})X_{99}^{(4)}(\rho _{2}+2-n_{1}, c+1) \\ &{}-\frac{y\rho _{1}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{1}+1, \rho _{2}+1-n_{1}, c+1), \end{aligned} \end{aligned}$$
(2.78)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{3}+n)=X_{99}^{(4)}+ \frac{u\rho _{4}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{4}+1, \rho _{3}+n_{1}, c+1), \end{aligned}$$
(2.79)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{3}-n)=X_{99}^{(4)}- \frac{u\rho _{4}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{4}+1, \rho _{3}+1-n_{1}, c+1), \end{aligned}$$
(2.80)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{4}+n)=X_{99}^{(4)}+ \frac{u\rho _{3}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{3}+1, \rho _{4}+n_{1}, c+1), \end{aligned}$$
(2.81)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{4}-n)=X_{99}^{(4)}- \frac{u\rho _{3}}{c}\sum_{n_{1}=1}^{n}X_{99}^{(4)}( \rho _{3}+1, \rho _{4}+1-n_{1}, c+1). \end{aligned}$$
(2.82)

Theorem 2.14

The following recursion formulas hold true for the numerator parameters \(\rho _{3}\), \(\rho _{4}\) of \(X_{99}^{(4)}\):

$$\begin{aligned}& X_{99}^{(4)}(\rho _{3}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} u^{n_{1}} }{(c)_{n_{1}}} X_{99}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, c+n_{1}), \end{aligned}$$
(2.83)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{3}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{4})_{n_{1}} (-u)^{n_{1}}}{(c)_{n_{1}}} X_{99}^{(4)}( \rho _{4}+n_{1}, c+n_{1}), \end{aligned}$$
(2.84)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{4}+n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} u^{n_{1}} }{(c)_{n_{1}}} X_{99}^{(4)}( \rho _{3}+n_{1}, \rho _{4}+n_{1}, c+n_{1}), \end{aligned}$$
(2.85)
$$\begin{aligned}& X_{99}^{(4)}(\rho _{4}-n)= \sum _{n_{1}\leq n}\binom{n}{n_{1}} \frac{(\rho _{3})_{n_{1}} (-u)^{n_{1}}}{(c)_{n_{1}}} X_{99}^{(4)}( \rho _{3}+n_{1}, c+n_{1}). \end{aligned}$$
(2.86)

Theorem 2.15

The following recursion formula holds true for the denominator parameter c of \(X_{99}^{(4)}\):

$$\begin{aligned}& X_{99}^{(4)}(\omega _{1}-n) \\& \quad = X_{99}^{(4)} + (\rho _{1})_{2} x\sum_{n_{1}=1}^{n} \frac{1}{(c - n_{1})(c+1-n_{1})} X_{99}^{(4)}(\rho _{1}+2, c+2-n_{1}) \\& \qquad {} + \rho _{1} \rho _{2} y\sum _{n_{1}=1}^{n} \frac{1}{(c - n_{1})(c+1-n_{1})} X_{99}^{(4)}(\rho _{1}+1, \rho _{2}+1, c+2-n_{1}) \\& \qquad {} + (\rho _{2})_{2} z\sum_{n_{1}=1}^{n} \frac{1}{(c - n_{1})(c+1-n_{1})} X_{99}^{(4)}(\rho _{2}+2, c+2-n_{1}) \\& \qquad {} + \rho _{3} \rho _{4} u\sum _{n_{1}=1}^{n} \frac{1}{(c - n_{1})(c+1-n_{1})} X_{99}^{(4)}(\rho _{3}+1, \rho _{4}+1, c+2-n_{1}). \end{aligned}$$
(2.87)

Conclusion

This paper presented recursion formulas for some hypergeometric functions of four variables. Also, some interesting particular cases and consequences of our results have been discussed. New structures of hypergeometric functions of four variables emerge with wide possibilities of application in physics and engineering within such a context. Therefore, the results of this work are various, significant, and it would be interesting and possible to develop this study in the future.

Availability of data and materials

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. 1.

    Agarwal, P.: Certain properties of the generalized Gauss hypergeometric functions. Appl. Math. Inf. Sci. 8(5), 2315–2320 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Agarwal, P., Chand, M., Purohit, S.D.: A note on generating functions involving the generalized Gauss hypergeometric functions. Nat. Acad. Sci. Lett. 37(5), 457–459 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Rahman, G., Nisar, K.S., Arshad, M.: A new extension and applications of Caputo fractional derivative operator. Analysis 41(1), 1–11 (2021)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Saboor, A., Rahman, G., Anjum, Z., Nisar, K.S., Araci, S.: A new extension of Srivastava’s triple hypergeometric functions and their associated properties. Analysis 41(1), 13–24 (2021)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Agarwal, P., Younis, J.A., Kim, T.: Certain generating functions for the quadruple hypergeometric series \(K_{10}\). Notes Number Theory Discrete Math. 25, 16–23 (2019)

    Article  Google Scholar 

  6. 6.

    Ali, R.S., Mubeen, S., Nayab, I., Araci, S., Rahman, G., Nisar, K.S.: Some fractional operators with the generalized Bessel–Maitland function. Discrete Dyn. Nat. Soc. 2020, Article ID 1378457 (2020)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Bin-Saad, M.G., Younis, J.A.: On generating functions of quadruple hypergeometric function \(X_{8}^{(4)}\). Turk. J. Anal. Number Theory 7, 5–10 (2019)

    Article  Google Scholar 

  8. 8.

    Brychkov, Y.A., Saad, S.: On some formulas for the Appell function \(F_{2}(a,b,b';c,c',w,z)\). Integral Transforms Spec. Funct. 25, 111–123 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hasanov, A., Saad, M.G.B., Ryskan, A.: Some properties of Horn type second order double hypergeometric series. Bull. KRASEC Phys. Math. Sci. 21, 32–47 (2018)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Rahman, G., Saboor, A., Anjum, Z., Nisar, K.S., Abdeljawad, T.: An extension of the Mittag-Leffler function and its associated properties. Adv. Math. Phys. 2020, Article ID 5792853 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Younis, J.A., Bin-Saad, M., Verma, A.: Generating functions for some hypergeometric functions of four variables via Laplace integral representations. J. Funct. Spaces 2021, Article ID 7638597 (2021)

    MathSciNet  Google Scholar 

  12. 12.

    Mohammed, P.O., Aydi, H., Kashuri, A., Hamed, Y.S., Abualnaja, K.M.: Midpoint inequalities in fractional calculus defined using positive weighted symmetry function kernels. Symmetry 13, 550 (2021)

    Article  Google Scholar 

  13. 13.

    Butt, S.I., Set, E., Yousaf, S., Abdeljawad, T., Shatanawi, W.: Generalized integral inequalities for ABK-fractional integral operators. AIMS Math. 6(9), 10164–10191 (2021)

    Article  Google Scholar 

  14. 14.

    Hasanov, A., Younis, J., Aydi, H.: On decomposition formulas related to the Gaussian hypergeometric functions in three variables. J. Funct. Spaces 2021, Article ID 5538583 (2021)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Set, E., Butt, S.I., Akdemir, A.O., Karaoglan, A., Abdeljawad, T.: New integral inequalities for differentiable convex functions via Atangana–Baleanu fractional integral operators. Chaos Solitons Fractals 143, 110554 (2021)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Younis, J.A., Aydi, H., Verma, A.: Some formulas for new quadruple hypergeometric functions. J. Math. 2021, Article ID 5596299 (2021)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Bell, W.W.: Special Functions for Scientists and Engineers. Oxford University press, London (1968)

    MATH  Google Scholar 

  18. 18.

    Frankl, F.I.: Selected Works in Gas Dynamics. Nauka, Moscow (1973) (in Russian)

    Google Scholar 

  19. 19.

    Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Ellis Horwood, Chichester (1984)

    MATH  Google Scholar 

  20. 20.

    Abdeljawad, T., Baleanu, D.: Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10, 1098–1107 (2017)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Exton, H.: Multiple Hypergeometric Functions and Applications. Halsted, New York (1976)

    MATH  Google Scholar 

  22. 22.

    Sharma, C., Parihar, C.L.: Hypergeometric functions of four variables (I). J. Indian Acad. Math. 11, 121–133 (1989)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Bin-Saad, M.G., Younis, J.A.: Certain quadruple hypergeometric series and their integral representations. Appl. Appl. Math. 14, 1085–1098 (2019)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Bin-Saad, M.G., Younis, J.A.: On connections between certain class of new quadruple and known triple hypergeometric series. Tamap J. Math. Stat. 2019, Article ID SI02 (2019)

    Google Scholar 

  25. 25.

    Sahai, V., Verma, A.: Recursion formulas for Exton’s triple hypergeometric functions. Kyungpook Math. J. 56, 473–506 (2016)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Opps, S.B., Saad, N., Srivastava, H.M.: Recursion formulas for Appell’s hypergeometric function \(F_{2}\) with some applications to radiation field problems. Appl. Math. Comput. 207, 545–558 (2009)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Wang, X.: Recursion formulas for Appell functions. Integral Transforms Spec. Funct. 23(6), 421–433 (2012)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Sahai, V., Verma, A.: Recursion formulas for multivariable hypergeometric functions. Asian-Eur. J. Math. 8(4), 1550082 (2015)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Sahai, V., Verma, A.: Recursion formulas for the Srivastava–Daoust and related multivariable hypergeometric functions. Asian-Eur. J. Math. 9(4), 1650081 (2016)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Sahai, V., Verma, A.: Recursion formulas for Srivastava’s general triple hypergeometric function. Asian-Eur. J. Math. 9(3), 1650063 (2016)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to the Deanship of Post Graduate and Scientific Research at Dar Al Uloom University for funding this work.

Funding

This work did not receive any external funding.

Author information

Affiliations

Authors

Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Habes Alsamir.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Younis, J., Verma, A., Aydi, H. et al. Recursion formulas for certain quadruple hypergeometric functions. Adv Differ Equ 2021, 407 (2021). https://doi.org/10.1186/s13662-021-03561-z

Download citation

MSC

  • 15A15
  • 33C65

Keywords

  • Recursion formula
  • Quadruple hypergeometric functions