In recent years, many researchers introduced and studied several extensions and generalizations of various special functions due to its applications in diverse areas of mathematical, physical, engineering, etc. Agarwal et al. [1,2] established some properties for generalized Gauss hypergeometric functions, which were introduced by Özergin et al. Later, Agarwal et al. [3] and Çetinkaya et al. [4] introduced and investigated further extensions of Appell’s hypergeometric functions of two variables and Lauricella’s hypergeometric functions of three variables by using the generalized Beta type function. Purohit et al. [5] investigated Chebyshev type inequalities involving fractional integral operator containing a multi-index Mittag-Leffler function in the kernel. Suthar et al. [6] introduced certain generalized forms of the fractional kinetic equation pertaining to the (p, q)-Mathieu-type power series using the Laplace transforms technique. Chandola et al. [7] defined a new extension of beta function using the Appell series and the Lauricella function. The interested reader may be referred to several recent papers on the subject (see, e.g., [8–11] and the references cited therein).

Hypergeometric functions in several variables have many applications in applied problems (see, e.g., [12–16]). Also, multidimensional hypergeometric functions are used to solve boundary value problems (Dirichlet problem, Neumann problem, Holmgren problem, etc) for multidimensional degenerate differential equations (see [17–19]). In [20], Exton defined twenty one complete hypergeometric functions in four variables denoted by the symbols K₁, K₂,…, K₂₁. In [21], Sharma et al. introduced eighty three complete quadruple hypergeometric functions, namely F1(4),F2(4),…,F83(4). Very recently, Younis et al. [22] introduced and studied further quadruple hypergeometric functions denoted by X85(4),X86(4),…,X90(4). Each quadruple hypergeometric function in [20–22] is of the form: X(4)(.)=∑m,n,p,q=0∞Δ(m,n,p,q)xmm!ynn!zpp!uqq!, where Δ(m, n, p, q) is a certain sequence of complex parameters and there are twelve parameters in each series X⁽⁴⁾(.) (eight a′s and four c′s). The 1st, 2nd, 3rd and 4th parameters in X⁽⁴⁾(.) are connected with the integers m, n, p and q, respectively. Each repeated parameter in the series X⁽⁴⁾(.) points out a term with double parameters in δ(m, n, p, q). For example, X⁽⁴⁾(σ1,σ1,σ2,σ2,σ3,σ3,σ4,σ5) mean that (σ1)m+n(σ2)p+q (σ3)m+n(σ4)p(σ5)_q includes the term. Similarly, X⁽⁴⁾(σ1,σ1,σ1,σ2,σ1,σ1,σ2,σ3) points out the term (σ1)2m+2n+p (σ2)p+q (σ3)_q and X⁽⁴⁾(σ1, σ1, σ2, σ4, σ1, σ2, σ3, σ5) shows the existence of the term (σ1)2m+n(σ2)n+p(σ3)_p(σ4)_q(σ5)q. Thus, it is possible to form various combinations of indices. There seems to be no way of establishing independently the number of distinct Gaussian hypergeometric series for any given integer n≥ 2 without stating explicitly all such series. Thus, in every situation with n = 4, one ought to begin by actually constructing the set just as in the case n = 3 (see [23]). Motivated by the works [20–22], we decide to define further hypergeometric functions in four variables as follows:

(1) X91(4)(σ1,σ1,σ2,σ4,σ1,σ2,σ3,σ5;ρ1,ρ1,ρ2,ρ1;x,y,z,u)=∑m,n,p,q=0∞(σ1)2m+n(σ2)n+p(σ3)p(σ4)q(σ5)q(ρ1)m+n+p(ρ2)qxmm!ynn!zpp!uqq!,(|x|<14,|y|<1,|z|<1,|u|<1);

(2) X92(4)(σ1,σ1,σ2,σ4,σ1,σ2,σ3,σ5;ρ1,ρ2,ρ1,ρ1;x,y,z,u)=∑m,n,p,q=0∞(σ1)2m+n(σ2)n+p(σ3)p(σ4)q(σ5)q(ρ1)m+p+q(ρ2)nxmm!ynn!zpp!uqq!,(|x|<14,|y|<1,|z|<1,|u|<1);

where (a)m:=Γ(a+m)Γ(a),(a+m∈C∖Z0−)={1(m=0)a(a+1)…(a+m−1)(m=n∈N).

Here, C,Z0− and N denote the sets of complex numbers, non-positive integers, and positive integers, respectively.

Recently, many authors have obtained several recursion formulas involving hypergeometric functions in several variables. In Opps et al. [24], introduced the recursion formulas for the Appell’s function F₂ and gave its applications to radiation field problems. Wang [25] presented the recursion formulas for Appell functions F₁, F₂, F₃ and F₄. Sahai et al. [26,27] 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. Shehata et al. [28] discussed and derived new recursion relations for the Horn’s hypergeometric functions. In this present paper, we aim to establish several recursion formulas for the new hypergeometric functions in four variables defined by (1.1)–(1.4).

The following abbreviated notations are used in this paper. We, for example, write X91(4) for the series X91(4)(σ1,σ1,σ2,σ4,σ1,σ2,σ3,σ5;ρ1,ρ1,ρ2,ρ1;x,y,z,u) and X91(4)(σ1+n) for X91(4)(σ1+n,σ1+n,σ2,σ4,σ1,σ2,σ3,σ5;ρ1,ρ1,ρ2,ρ1;x,y,z,u). The notation X91(4)(σ1+n,σ2+n1) stands for X91(4)(σ1+n,σ1+n,σ2+n1,σ4,σ1+n,σ2+n1,σ3,σ5;ρ1,ρ1,ρ2,ρ1;x,y,z,u) and X91(4)(σ1+n,σ2+n1,ρ1+n2) stands for X91(4)(a1+n,σ1+n,σ2+n1,σ4,σ1+n,σ2+n1,σ3,σ5;ρ1+n2,ρ1+n2,ρ2,ρ1;x,y,z,u), etc.

Here, we establish several recursion formulas for our hypergeometric functions in four variables.

Theorem 2.1 The following recursion formulas hold true for the numerator parameter σ1, σ2, σ3, σ4, σ5 of the X91(4):

(5) X91(4)(σ1+n)=X91(4)+2xρ1∑n1=1n(σ1+n1)X91(4)(σ1+1+n1,ρ1+1)+yσ2ρ1∑n1=1nX91(4)(σ1+n1,σ2+1,ρ1+1),

(6) X91(4)(σ1−n)=X91(4)−2xρ1∑n1=1n(σ1+1−n1)X91(4)(σ1+2−n1,ρ1+1)−yσ2ρ1∑n1=1nX91(4)(σ1+1−n1,σ2+1,ρ1+1),

Proof. From the definition of the hypergeometric function X91(4) and the relation

(15) (σ1+1)2m+n=(σ1)2m+n(1+2mσ1+nσ1)

we obtain the following contiguous relation:

(16) X91(4)(σ1+1)=X91(4)+2xρ1(σ1+1)X91(4)(σ1+2,ρ1+1)+yσ2ρ1X91(4)(σ1+1,σ2+1,ρ1+1).

To find a contiguous relation for X91(4)(σ1+2), we replace σ1 by σ1 + 1 in (16) and simplify. This leads to:

(17) X91(4)(σ1+2)=X91(4)+2xρ1∑n1=12(σ1+n1)X91(4)(σ1+n1+1,ρ1+1)+yσ2ρ1∑n1=12X91(4)(σ1+n1,σ2+1,ρ1+1).

Iterating this process n-times, we obtain (5). For the proof of (6), replace the parameter σ1 by σ1−1 in (15). This implies that

(18) X91(4)(σ1−1)=X1−2xρ1σ1X91(4)(σ1+1,ρ1+1)−yσ2ρ1X91(4)(σ2+1,ρ1+1).

Iteratively, we get (6).

The recursion formulas from (7)–(14) can be proved in a similar manner.

Theorem 2.2 The following recursion formulas hold true for the numerator parameter σ2, σ3, σ4, σ5 of the X91(4):

(19) X91(4)(σ2+n)=∑N2≤n(nn1,n2)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2),

(20) X91(4)(σ2−n)=∑N2≤n(nn1,n2)(σ1)n1(σ3)n2(−y)n1(−z)n2(ρ1)n1(ρ2)n2X91(4)(σ1+n1,σ3+n2,ρ1+n1,ρ2+n2),

where (nn1,n2)=n!n1!n2!(n−n1−n2)! and N2=n1+n2.

Proof. The proof of (19) is based upon the principle of a mathematical induction on n∈N. For n = 1, the result (19) is true obviously following (7). Suppose (19) is true for n = m, that is,

(27) X91(4)(σ2+m)=∑N2≤m(nn1,n2)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2),

Replacing σ2 with σ2+ 1 in (27) and using the contiguous relation (7) for n = 1, we get

(28) X91(4)(σ2+m+1)=∑N2≤m(nn1,n2)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2×{X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2)+(σ1+n1)y(ρ1+n1)X91(4)(σ1+n1+1,σ2+N2+1,σ3+n2,ρ1+n1+1,ρ2+n1)+(σ3+n2)z(ρ2+n2)X91(4)(σ1+n1,σ2+N2+1,σ3+n2+1,ρ1+n1,ρ2+n2+1)}.

By a simplification, (28) takes the form

(29) X91(4)(a2+m+1)=∑N2≤m(nn1,n2)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2×X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2)+∑N2≤m+1(nn1−1,n2)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2×X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2)+∑N2≤m+1(nn1,n2−1)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2×X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2).

Using the Pascal’s identity in (29), we have X91(4)(σ2+m+1)=∑N2≤m+1(nn1,n2)(σ1)n1(σ3)n2yn1zn2(ρ1)n1(ρ2)n2×X91(4)(σ1+n1,σ2+N2,σ3+n2,ρ1+n1,ρ2+n2).

This establishes (19) for n = m + 1. Hence, by induction, the result given in (19) is true for all values of n. The recursion formulas (20)–(26) can be proved in a similar manner.

Theorem 2.3 The following recursion formulas hold true for the denominator parameter ρ1, ρ2 of the X91(4):

(30) X91(4)(ρ1−n)=X91(4)+(σ1)2x∑n1=1n1(ρ1−n1)(ρ1+1−n1)X91(4)(σ1+2,ρ1+2−n1)+σ1σ2y∑n1=1n1(ρ1−n1)(ρ1+1−n1)X91(4)(σ1+1,σ2+1,ρ1+2−n1)+σ4σ5u∑n1=1n1(ρ1−n1)(ρ1+1−n1)X91(4)(σ4+1,σ5+1,ρ1+2−n1),

(31) X91(4)(ρ2−n)=X91(4)+σ4σ5u∑n1=1n1(ρ2−n1)(ρ2+1−n1)X91(4)(σ4+1,σ5+1,ρ2+2−n1).

Proof. Applying the definition of the hypergeometric function X91(4) and the relation

(32) 1(ρ1−1)m+n+q=1(ρ1)m+n+q(1+mρ1−1+nρ1−1+qρ1−1),

we have:

(33) X91(4)(ρ1−1)=X91(4)+(σ1)2xρ1(ρ1−1)X91(4)(σ1+2,ρ1+1)+σ1σ2yρ1(ρ1−1)X91(4)(σ1+1,σ2+1,ρ1+2−n1)+σ4σ5uρ1(ρ1−1)X91(4)(σ4+1,σ5+1,ρ1+2−n1).

Using this contiguous relation to the X91(4) with the parameter ρ1−n for n-times, we get the result (30). The recursion formula (31) can be proved in a similar manner.

Theorem 2.4 The following recursion formulas hold true for the numerator parameter σ1, σ2, σ3, σ4, σ5 of the X92(4):

(34) X92(4)(σ1+n)=X92(4)+2xρ1∑n1=1n(σ1+n1)X92(4)(σ1+1+n1,ρ1+1)+yσ2ρ2∑n1=1nX92(4)(σ1+n1,σ2+1,ρ2+1),

(35) X92(4)(σ1−n)=X92(4)−2xρ1∑n1=1n(σ1+1−n1)X92(4)(σ1+2−n1,ρ1+1)−yσ2ρ2∑n1=1nX92(4)(σ1+1−n1,σ2+1,ρ2+1),