Some relations following from the decomposition formula for one   multidimensional Lauricella hypergeometric function

Tuhtasin Ergashev

arXiv:1905.03962·math.AP·May 13, 2019

Some relations following from the decomposition formula for one multidimensional Lauricella hypergeometric function

Tuhtasin Ergashev

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TL;DR

This paper explores relations derived from the decomposition formula of a multidimensional Lauricella hypergeometric function, aiding the analysis of fundamental solutions for certain elliptic equations with singular coefficients.

Contribution

It identifies specific relations from the decomposition formula of a Lauricella hypergeometric function, advancing understanding of multidimensional hypergeometric functions.

Findings

01

Derived relations from the decomposition formula for Lauricella hypergeometric functions.

02

Enhanced methods for analyzing fundamental solutions of elliptic equations with singular coefficients.

03

Contributed to the theoretical framework connecting hypergeometric functions and elliptic PDEs.

Abstract

Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, some relations following from the decomposition formula for one multidimensional Lauricell hypergeometric function are determined.

Equations126

L_{α}^{(m, n)} (u) := i = 1 \sum m u_{x_{i} x_{i}} + j = 1 \sum n \frac{2 α _{j}}{x _{j}} u_{x_{j}} = 0

L_{α}^{(m, n)} (u) := i = 1 \sum m u_{x_{i} x_{i}} + j = 1 \sum n \frac{2 α _{j}}{x _{j}} u_{x_{j}} = 0

F_{A}^{(n)} (a, b_{1}, ..., b_{n}; c_{1}, ..., c_{n}; z_{1}, ..., z_{n})

F_{A}^{(n)} (a, b_{1}, ..., b_{n}; c_{1}, ..., c_{n}; z_{1}, ..., z_{n})

F_{A}^{\left({n}\right)}\left({a,b_{1},...,b_{n};c_{1},...,c_{n};z_{1},...,z_{n}}\right)=F_{A}^{(n)}{\left[{{\begin{array}[]{*{20}c}{a,b_{1},...,b_{n}\hfill}\\ {c_{1},...,c_{n}\hfill}\\ \end{array}}z_{1},...,z_{n}}\right]}

F_{A}^{\left({n}\right)}\left({a,b_{1},...,b_{n};c_{1},...,c_{n};z_{1},...,z_{n}}\right)=F_{A}^{(n)}{\left[{{\begin{array}[]{*{20}c}{a,b_{1},...,b_{n}\hfill}\\ {c_{1},...,c_{n}\hfill}\\ \end{array}}z_{1},...,z_{n}}\right]}

= p_{1}, ..., p_{n} = 0 \sum \infty \frac{( a ) _{p_{1} + ... + p_{n}} ( b _{1} ) _{p_{1}} ... ( b _{n} ) _{p_{n}}}{( c _{1} ) _{p_{1}} ... ( c _{n} ) _{p_{n}}} \frac{z _{1}^{p_{1}}}{p _{1} !} ... \frac{z _{n}^{p_{n}}}{p _{n} !}, i = 1 \sum n ∣ z_{i} ∣ < 1,

= p_{1}, ..., p_{n} = 0 \sum \infty \frac{( a ) _{p_{1} + ... + p_{n}} ( b _{1} ) _{p_{1}} ... ( b _{n} ) _{p_{n}}}{( c _{1} ) _{p_{1}} ... ( c _{n} ) _{p_{n}}} \frac{z _{1}^{p_{1}}}{p _{1} !} ... \frac{z _{n}^{p_{n}}}{p _{n} !}, i = 1 \sum n ∣ z_{i} ∣ < 1,

(1)_{m} = m! (m \in N_{0} := N \cup {0}; N = {1, 2, 3, ...}),

(1)_{m} = m! (m \in N_{0} := N \cup {0}; N = {1, 2, 3, ...}),

(κ)_{0} = 1, (κ)_{ν} = \frac{Γ ( κ + ν )}{Γ ( κ )}, ν \in N .

(κ)_{0} = 1, (κ)_{ν} = \frac{Γ ( κ + ν )}{Γ ( κ )}, ν \in N .

\begin{array}[]{l}q_{k}\left({x,\xi}\right)=\gamma_{k}{\prod\limits_{i=1}^{k}{\left({x_{i}\xi_{i}}\right)^{1-2\alpha_{i}}\cdot}}r^{-2\bar{\alpha}_{k}}F_{A}^{\left({n}\right)}{\left[{{\begin{array}[]{*{20}c}{\bar{\alpha}_{k},1-\alpha_{1},...,1-\alpha_{k},\alpha_{k+1},...,\alpha_{n};\hfill}\\ {2-2\alpha_{1},...,2-2\alpha_{k},2\alpha_{k+1},...,2\alpha_{n};\hfill}\\ \end{array}}\sigma_{1},...,\sigma_{n}}\right]},\\ \end{array}

\begin{array}[]{l}q_{k}\left({x,\xi}\right)=\gamma_{k}{\prod\limits_{i=1}^{k}{\left({x_{i}\xi_{i}}\right)^{1-2\alpha_{i}}\cdot}}r^{-2\bar{\alpha}_{k}}F_{A}^{\left({n}\right)}{\left[{{\begin{array}[]{*{20}c}{\bar{\alpha}_{k},1-\alpha_{1},...,1-\alpha_{k},\alpha_{k+1},...,\alpha_{n};\hfill}\\ {2-2\alpha_{1},...,2-2\alpha_{k},2\alpha_{k+1},...,2\alpha_{n};\hfill}\\ \end{array}}\sigma_{1},...,\sigma_{n}}\right]},\\ \end{array}

\overset{α}{ˉ}_{k} = \frac{m}{2} + k - 1 - i = 1 \sum k α_{i} + i = k + 1 \sum n α_{i},

\overset{α}{ˉ}_{k} = \frac{m}{2} + k - 1 - i = 1 \sum k α_{i} + i = k + 1 \sum n α_{i},

γ_{k} = 2^{2 \overset{α}{ˉ}_{k} - m} \frac{Γ ( α ˉ _{k} )}{π ^{m /2}} i = k + 1 \prod n \frac{Γ ( α _{i} )}{Γ ( 2 α _{i} )} j = 1 \prod k \frac{Γ ( 1 - α _{j} )}{Γ ( 2 - 2 α _{j} )}, k = \overline{0, n};

γ_{k} = 2^{2 \overset{α}{ˉ}_{k} - m} \frac{Γ ( α ˉ _{k} )}{π ^{m /2}} i = k + 1 \prod n \frac{Γ ( α _{i} )}{Γ ( 2 α _{i} )} j = 1 \prod k \frac{Γ ( 1 - α _{j} )}{Γ ( 2 - 2 α _{j} )}, k = \overline{0, n};

σ_{k} = 1 - \frac{r _{k}^{2}}{r ^{2}}, r^{2} = i = 1 \sum m (x_{i} - ξ_{i})^{2}, r_{k}^{2} = (x_{k} + ξ_{k})^{2} + i = 1, i \neq = k \sum m (x_{i} - ξ_{i})^{2}, k = \overline{1, n} .

σ_{k} = 1 - \frac{r _{k}^{2}}{r ^{2}}, r^{2} = i = 1 \sum m (x_{i} - ξ_{i})^{2}, r_{k}^{2} = (x_{k} + ξ_{k})^{2} + i = 1, i \neq = k \sum m (x_{i} - ξ_{i})^{2}, k = \overline{1, n} .

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y) = i, j = 0 \sum \infty \frac{( a ) _{i + j} ( b _{1} ) _{i} ( b _{2} ) _{j}}{( c _{1} ) _{i} ( c _{2} ) _{j}} \frac{x ^{i}}{i !} \frac{y ^{j}}{j !}

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y) = i, j = 0 \sum \infty \frac{( a ) _{i + j} ( b _{1} ) _{i} ( b _{2} ) _{j}}{( c _{1} ) _{i} ( c _{2} ) _{j}} \frac{x ^{i}}{i !} \frac{y ^{j}}{j !}

[c_{1}, c_{2} \neq = 0, - 1, - 2, ...; ∣ x ∣ + ∣ y ∣ < 1]

[c_{1}, c_{2} \neq = 0, - 1, - 2, ...; ∣ x ∣ + ∣ y ∣ < 1]

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y)

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y)

= i = 0 \sum \infty \frac{( a ) _{i} ( b _{1} ) _{i} ( b _{2} ) _{i}}{i ! ( c _{1} ) _{i} ( c _{2} ) _{i}} x^{i} y^{i} F (a + i, b_{1} + i; c_{1} + i; x) F (a + i, b_{2} + i; c_{2} + i; y),

= i = 0 \sum \infty \frac{( a ) _{i} ( b _{1} ) _{i} ( b _{2} ) _{i}}{i ! ( c _{1} ) _{i} ( c _{2} ) _{i}} x^{i} y^{i} F (a + i, b_{1} + i; c_{1} + i; x) F (a + i, b_{2} + i; c_{2} + i; y),

F{\left(a,b;c;z\right)}=F{\left[{{\begin{array}[]{*{20}c}{a,b};\hfill\\ {c};\hfill\\ \end{array}}z}\right]}={\sum\limits_{i=0}^{\infty}\frac{(a)_{i}(b)_{i}}{(c)_{i}i!}z^{i}}

F{\left(a,b;c;z\right)}=F{\left[{{\begin{array}[]{*{20}c}{a,b};\hfill\\ {c};\hfill\\ \end{array}}z}\right]}={\sum\limits_{i=0}^{\infty}\frac{(a)_{i}(b)_{i}}{(c)_{i}i!}z^{i}}

\nabla (h) = \frac{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}, Δ (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )},

\nabla (h) = \frac{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}, Δ (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )},

\tilde{\nabla}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )},

\tilde{\nabla}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )},

\tilde{Δ}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )}{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )},

\tilde{Δ}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )}{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )},

F_{A}^{(n)} (a, b_{1}, ..., b_{n}; c_{1}, ..., c_{n}; z_{1}, ..., z_{n})

F_{A}^{(n)} (a, b_{1}, ..., b_{n}; c_{1}, ..., c_{n}; z_{1}, ..., z_{n})

= m_{2}, ..., m_{n} = 0 \sum \infty \frac{( a ) _{m_{2} + ... + m_{n}} ( b _{1} ) _{m_{2} + ... + m_{n}} ( b _{2} ) _{m_{2}} ... ( b _{n} ) _{m_{n}}}{m _{2} ! ... m _{n} ! ( c _{1} ) _{m_{2} + ... + m_{n}} ( c _{2} ) _{m_{2}} ... ( c _{n} ) _{m_{n}}} z_{1}^{m_{2} + ... + m_{n}} z_{2}^{m_{2}} ... z_{n}^{m_{n}}

= m_{2}, ..., m_{n} = 0 \sum \infty \frac{( a ) _{m_{2} + ... + m_{n}} ( b _{1} ) _{m_{2} + ... + m_{n}} ( b _{2} ) _{m_{2}} ... ( b _{n} ) _{m_{n}}}{m _{2} ! ... m _{n} ! ( c _{1} ) _{m_{2} + ... + m_{n}} ( c _{2} ) _{m_{2}} ... ( c _{n} ) _{m_{n}}} z_{1}^{m_{2} + ... + m_{n}} z_{2}^{m_{2}} ... z_{n}^{m_{n}}

\cdot F (a + m_{2} + ... + m_{n}, b_{1} + m_{2} + ... + m_{n}; c_{1} + m_{2} + ... + m_{n}; z_{1})

\cdot F (a + m_{2} + ... + m_{n}, b_{1} + m_{2} + ... + m_{n}; c_{1} + m_{2} + ... + m_{n}; z_{1})

\cdot F_{A}^{(n - 1)} (a + m_{2} + ... + m_{n}, b_{2} + m_{2}, ..., b_{n} + m_{n};

\cdot F_{A}^{(n - 1)} (a + m_{2} + ... + m_{n}, b_{2} + m_{2}, ..., b_{n} + m_{n};

c_{2} + m_{2}, ..., c_{n} + m_{n}; z_{2}, ..., z_{n}), n \in N \ {1} .

c_{2} + m_{2}, ..., c_{n} + m_{n}; z_{2}, ..., z_{n}), n \in N \ {1} .

F_{A}^{(n)} (a, b_{1}, b_{2}, ...., b_{n}; c_{1}, c_{2}, ...., c_{n}; z_{1}, ..., z_{n})

F_{A}^{(n)} (a, b_{1}, b_{2}, ...., b_{n}; c_{1}, c_{2}, ...., c_{n}; z_{1}, ..., z_{n})

= (2 \leq i \leq j \leq n) m_{i, j} = 0 \sum \infty \frac{( a ) _{A (n, n)}}{m _{ij} !} k = 1 \prod n [\frac{( b _{k} ) _{B (k, n)}}{( c _{k} ) _{B (k, n)}} z_{k}^{B (k, n)} F (a + A (k, n),

= (2 \leq i \leq j \leq n) m_{i, j} = 0 \sum \infty \frac{( a ) _{A (n, n)}}{m _{ij} !} k = 1 \prod n [\frac{( b _{k} ) _{B (k, n)}}{( c _{k} ) _{B (k, n)}} z_{k}^{B (k, n)} F (a + A (k, n),

b_{k} + B (k, n); c_{k} + B (k, n); z_{k})],

b_{k} + B (k, n); c_{k} + B (k, n); z_{k})],

A (k, n) = i = 2 \sum k + 1 j = i \sum n m_{i, j}, B (k, n) = i = 2 \sum k m_{i, k} + i = k + 1 \sum n m_{k + 1, i} .

A (k, n) = i = 2 \sum k + 1 j = i \sum n m_{i, j}, B (k, n) = i = 2 \sum k m_{i, k} + i = k + 1 \sum n m_{k + 1, i} .

k = 2 \sum n i = 2 \sum k m_{i, k} = k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i}

k = 2 \sum n i = 2 \sum k m_{i, k} = k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i}

k = 1 \sum n B (k, n) = 2 k = 2 \sum n i = 2 \sum k m_{i, k} = 2 k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i} .

k = 1 \sum n B (k, n) = 2 k = 2 \sum n i = 2 \sum k m_{i, k} = 2 k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i} .

A (n + 1, n + 1) - B (n + 1, n + 1) = A (n, n),

A (n + 1, n + 1) - B (n + 1, n + 1) = A (n, n),

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TopicsMathematical functions and polynomials · Iterative Methods for Nonlinear Equations · Diverse Research Studies Overview

Full text

**Some relations following from the decomposition formula

for one multidimensional Lauricella hypergeometric function

** **Ergashev T.G.

** Institute of Mathematics, Uzbek Academy of Sciences, Tashkent, Uzbekistan.

[email protected]

Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, some relations following from the decomposition formula for one multidimensional Lauricell hypergeometric function are determined.

Key words: multidimensional elliptic equation with several singular coefficients; fundamental solutions; multiple Lauricella hypergeometric functions; decomposition formula; summation formula.

We consider the equation

[TABLE]

in the region $R_{m}^{n+}={\left\{{x:x_{1}>0,x_{2}>0,...,x_{n}>0}\right\}},$ where $x=\left({x_{1},...,x_{m}}\right),\,m\geq 2,\\ 0\leq n\leq m;\alpha=\left({\alpha_{1},...,\alpha_{n}}\right),\alpha_{j}$ are real numbers with $0<2\alpha_{j}<1,j=\overline{1,n}.$

Fundamental solutions of the equation were constructed recently [4]. In fact, the fundamental solutions of the equation (1) can be expressed in terms of Lauricella’s hypergeometric function in $n\,$ variables, that is, the Lauricella multivariable hypergeometric function

[TABLE]

defined by

[TABLE]

where $c_{i}\neq 0,-1,-2,...,\,\,i=\overline{1,n}$ and $\left({\kappa}\right)_{\nu}$ denotes the general Pochhammer symbol or the shifted factorial, since

[TABLE]

which is defined in terms of the familiar Gamma function, by

[TABLE]

We thus obtain the following fundamental solutions:

[TABLE]

where

[TABLE]

Here ${\sum\limits_{i=l+1}^{l}{}}$ is to be interpreted as zero, when $l=0$ or $l=n$ , and ${\prod\limits_{i=l+1}^{l}{}}$ is to be interpreted as one, when $l=0$ or $l=n$ .

For a given multiple hypergeometric function, it is useful to fund a decomposition formula which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. Burchnall and Chaundy [1, 2] systematically presented a number of expansion and decomposition formulas for some double hypergeometric functions in series of simpler hypergeometric functions. For example, the Appell function

[TABLE]

has the expansion [1]

[TABLE]

where

[TABLE]

is Gaussian hypergeometric function [3].

The Burchnall-Chaundy method, which is limited to functions of two variables, is based on the following mutually inverse symbolic operators [1]

[TABLE]

where ${\rm\delta}_{1}=x{\displaystyle\frac{{\partial}}{{\partial x}}}$ and ${\rm\delta}_{2}=y{\displaystyle\frac{{\partial}}{{\partial y}}}$ .

In order to generalize the operators $\nabla\left({h}\right)$ and $\Delta\left({h}\right)$ , defined in (4), A.Hasanov and H.M.Srivastava [5, 6] introduced the operators

[TABLE]

where ${\rm\delta}_{k}=z_{k}{\displaystyle\frac{{\partial}}{{\partial z_{k}}}}\,,$ with the help of which they managed to find decomposition formulas for a whole class of hypergeometric functions in several variables. For example, the hypergeometric Lauricella function $F_{A}^{\left({n}\right)}$ , defined by formula (2) has the decomposition formula [5]

[TABLE]

However, due to the recurrence of formula (7), additional difficulties may arise in the applications of this expansion. Further study of the properties of operators (5) and (6) showed that formula (7) can be reduced to a more convenient form.

Lemma 1 [4]. The following decomposition formula holds true at $n\in{\rm N}\backslash{\left\{{1}\right\}}$

[TABLE]

where

[TABLE]

The formula (8) is proved by the method mathematical induction [4].

It should be noted here that the sum ${\sum\limits_{k=1}^{n}{B(k,n)}}$ has the parity property, which plays an important role in the calculation of the some values of hypergeometric functions. In fact, by virtue of equality

[TABLE]

we obtain

[TABLE]

We present some simple properties of the functions $A\left({k,n}\right)$ and $B\left({k,n}\right)$ , defined by the formula (9):

[TABLE]

Those properties are easily proved if we proceed from the definitions of functions $A\left({k,n}\right)$ and $B\left({k,n}\right)$ .

Lemma 2. Let $a,b_{1},$ …, $b_{n}$ are real numbers with $a=0,\,-1,\,-2,...$ and $a>b_{1}+...+b_{n}.$ Then the following summation formula holds true at $n\in{\rm N}\backslash\{1\}$

[TABLE]

Note that if we put $n=2$ in the formula (14), then

[TABLE]

that is, the formula (13) is a natural generalization of the well-known summation formula for the Gauss hypergeometric function.

The proof of Lemma 2 is carried out by the method of mathematical induction.

From equality (14) it follows that the formula (13) is valid for $n=2$ .

Now we denote the left side of the formula (13) by

[TABLE]

and considering fair equality

[TABLE]

we will prove that

[TABLE]

For this aim we will put

[TABLE]

and show the validity of the recurrence relation

[TABLE]

This process consists of $n$ steps. A detailed look at the first step.

By virtue of the equalities

[TABLE]

and the properties of functions $A\left({k,n}\right)$ and $B\left({k,n}\right)$ (see formulas (11) and (12)), the right side of equality

[TABLE]

it is easy to convert to the form

[TABLE]

where

[TABLE]

It is easy to notice that

[TABLE]

Applying now the summation formula (15) to the last equality after elementary transformations we get

[TABLE]

For definiteness, we denote the result of the first step of the process under consideration by $T_{n+1}^{(\ref{eq1})}\left({a,b_{1},...,b_{n+1}}\right)$ . We continue the process of proving the recurrence relation (17). In each next step, having consistently repeated the reasoning carried out in the first step, we get

[TABLE]

and in the last step

[TABLE]

that is

[TABLE]

Thus, the validity of the ratio (17) is established. By the induction hypothesis, from the (17) follows the equality

[TABLE]

Substituting the last expression in (17) we get equality (14). Q.E.D.

Lemma 3. The following equality

[TABLE]

is valid.

Proof. By virtue of the decomposition formula (8) we obtain

[TABLE]

Applying now the familiar autotransformation formula

[TABLE]

to each hypergeometric function included in the sum (19), we get

[TABLE]

Next, we calculate the limit

[TABLE]

and the resulting expression we apply the summation formula (15), with the result that we obtain the equality (18). Q.E.D.

References

Список литературы

[1] Burchnall J.L., Chaundy T.W., Expansions of Appell’s double hypergeometric functions. The Quarterly Journal of Mathematics, Oxford, Ser.11,1940. 249-270.
[2] Burchnall J.L., Chaundy T.W., Expansions of Appell’s double hypergeometric functions(II). The Quarterly Journal of Mathematics, Oxford, Ser.12,1941. 112-128.
[3] Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F.G., Higher Transcendental Functions, Vol.I (New York, Toronto and London:McGraw-Hill Book Company), 1953.
[4] Ergashev T.G. Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. ArXiv.org.:1805.03826.
[5] Hasanov A., Srivastava H., Some decomposition formulas associated with the Lauricella function $F_{A}^{(r)}$ and other multiple hypergeometric functions, Applied Mathematic Letters, 19(2) (2006), 113-121.
[6] Hasanov A., Srivastava H., Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions, Computers and Mathematics with Applications, 53:7 (2007), 1119-1128.

Bibliography6

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Burchnall J.L., Chaundy T.W., Expansions of Appell’s double hypergeometric functions. The Quarterly Journal of Mathematics, Oxford, Ser.11,1940. 249-270.
2[2] Burchnall J.L., Chaundy T.W., Expansions of Appell’s double hypergeometric functions(II). The Quarterly Journal of Mathematics, Oxford, Ser.12,1941. 112-128.
3[3] Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F.G., Higher Transcendental Functions, Vol.I (New York, Toronto and London:Mc Graw-Hill Book Company), 1953.
4[4] Ergashev T.G. Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. Ar Xiv.org.:1805.03826.
5[5] Hasanov A., Srivastava H., Some decomposition formulas associated with the Lauricella function F A ( r ) superscript subscript 𝐹 𝐴 𝑟 F_{A}^{(r)} and other multiple hypergeometric functions, Applied Mathematic Letters, 19(2) (2006), 113-121.
6[6] Hasanov A., Srivastava H., Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions, Computers and Mathematics with Applications, 53:7 (2007), 1119-1128.