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Some Reduction Formulae Associated with Gauss and Fox-Wright Hypergeometric Functions

M.I. Qureshi, Saima Jabee*, Sulakshana Bajaj

Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi-110025, India

Abstract: In this paper, we describe some reduction formulae for Gauss’ hypergeometric function and Fox-Wright hypergeometric function associated with suitable convergence conditions using series rearrangement technique.

Keywords: Generalized hypergeometric function, Fox-Wright generalized hypergeometric function, Pfaff-Kummer’s linear transformation

Received January 20, 2018; Revised March 06, 2018; Accepted June 22, 2018

Citation: M.I. Qureshi, Saima Jabee, Sulakshana Bajaj, Some Reduction Formulae Associated with Gauss and Fox-Wright Hypergeometric Functions. Turkish Journal of Analysis and Number Theory, 6 (3). 103-106. January 2018. doi:10.12691/tjant-6-3-7

1. Introduction and Basic Notations

In the present paper, we shall use the following standard notations:

and .

Here, as usual, denotes the set of integers, denotes the set of real numbers, denotes the set of positive real numbers and denotes the set of complex numbers.

The Pochhammer symbol (or the shifted factorial) is defined in terms of the familiar Gamma function by

[picture] | (1.1)

it being understood conventionally that , and assumed tacitly that the Gamma quotient exists.

In the Gaussian hypergeometric series there are two numerator parameters , and one denominator parameter . A natural generalization of this series is accomplished by introducing any arbitrary number of numerator and denominator parameters. The non-terminating hypergeometric series [[5], pp.42-43]

[picture] | (1.2)

is known as the generalized Gauss and Kummer series, or simply, generalized hypergeometric series. Here and are positive integers or zero (interpreting an empty product as unity), and we assume that the variable , the numerator parameters and the denominator parameters take on complex values, provided that

[picture] | (1.3)

Convergence conditions [[5], p.43]:

Suppose that none of the numerator parameters is zero or negative integer (otherwise the question of convergence will not arise), and with the usual restriction (1.3), the series in the definition (1.2)

(i) converges for , if

(ii) converges for , if .

Furthermore, if we denote

it is known that the series, with , is

(a) absolutely convergent for , if ,

(b) conditionally convergent for if

Fox-Wright generalized hypergeometric function of one variable:

The Fox-Wright function of one variable ([[3], p.389]; see also [4,6,7]) is given by

[picture] | (1.4)

[picture] | (1.5)

[picture] | (1.6)

whereparameters coefficients in case of series (1.4) (or in case of contour integral (1.6)), In equation (1.4), the parameters and coefficients are adjusted in such a way that the product of Gamma functions in numerator and denominator should be well defined [1,2].

[picture] | (1.7)

[picture] | (1.8)

[picture] | (1.9)

[picture] | (1.10)

Case(I): When contour (L) is a left loop beginning and ending at , then given by (1.4) or (1.6) holds the following convergence conditions

i) when

ii) when

iii) when

Case(II): When contour (L) is a right loop beginning and ending at , then given by (1.4) or (1.6) holds the following convergence conditions

i) when

ii) when

iii) when

Case (III): When contour (L) is starting from and ending at , where , then is also convergent under the following conditions

i) when

ii) when such that

iii) when such that,

Next we collect some results that we will need in the sequel.

Decomposition identity:

The idea of separation of a power series into its even and odd terms exhibited by the elementary identity

[picture] | (1.11)

is atleast as old as the series themselves and concerned power series is absolutely convergent.

Pfaff-Kummer’s linear transformation [[5],p.33(eq.19)]:

[picture] | (1.12)

( and ).

Binomial theorem:

[picture] | (1.13)

; ,

In sections 2 and 3, we obtain some reduction formulae for Gauss’ hypergeometric function and Fox-Wright hypergeometric function associated with suitable convergence conditions by using binomial theorem, decomposition identity and Pfaff-Kummer’s linear transformation.

2. Some Reduction Formulae for Gauss’ Hypergeometric Function

The following reduction formulae associated with suitable convergence conditions hold true

[picture] | (2.1)

[picture] | (2.2)

[picture] | (2.3)

[picture] | (2.4)

Proof of reduction formula (2.1):

Suppose right hand side of equation (2.1) is denoted by . Then by using binomial theorem (1.13), we obtain

[picture] | (2.5)

Now using decomposition identity (1.11) in equation (2.5), we get

[picture] | (2.6)

Further using Pfaff-Kummer’s linear transformation (1.12) in equation (2.6), and simplifying further, we arrive at left hand side of equation (2.1).

Proof of reduction formula (2.2):

Suppose right hand side of equation (2.2) is denoted by . Then by using binomial theorem (1.13), we get

[picture] | (2.7)

Now using decomposition identity (1.11) in equation (2.7), we get

[picture] | (2.8)

On using Pfaff-Kummer’s linear transformation (1.12) in equation (2.8), and simplifying further, we have left hand side of equation (2.2).

Proof of reduction formula (2.3):

Suppose right hand side of equation (2.3) is denoted by . Then by using binomial theorem (1.13), we have

[picture] | (2.9)

Now using decomposition identity (1.11) in equation (2.9), we get

[picture] | (2.10)

On using Pfaff-Kummer’s linear transformation (1.12) in equation (2.10), and simplifying further, we arrive at left hand side of equation (2.3).

Proof of reduction formula (2.4):

Suppose right hand side of equation (2.4) is denoted by . Then by using binomial theorem (1.13), we obtain

[picture] | (2.11)

Now using decomposition identity (1.11) in equation (2.11), we get

[picture] | (2.12)

Applying Pfaff-Kummer’s linear transformation (1.12) in equation (2.12), and simplifying further, we get left hand side of equation (2.4).

3. Applications of Reduction Formulae in Fox-Wright function

As an application of formulae (2.1)-(2.4), we obtain the following two formulae for Fox-Wright hypergeometric function associated with convergence conditions

[picture] | (3.1)

and

[picture] | (3.2)

Proof of reduction formula (3.1):

Suppose left hand side of equation (3.1) is denoted by . Then by using the definitions of (1.4) and (1.5), we obtain

[picture] | (3.3)

Now using decomposition identity (1.11) in equation (3.3), we have

Solving further, we obtain

[picture] | (3.4)

Now applying the reduction formulae (2.1), (2.4), and simplifying further, we get the right hand side of equation (3.1).

Proof of reduction formula (3.2):

Suppose left hand side of equation (3.2) is denoted by . Then by using the definitions of (1.4) and (1.5), we get

[picture] | (3.5)

On using decomposition identity (1.11) in equation (3.5), we get

Solving further, we get

[picture] | (3.6)

Now applying the reduction formulae (2.2), (2.3), and solving further, we obtain the right hand side of equation (3.2).

Acknowledgements

The Authors are thankful to the anonymous referee for his/her valuable remarks and suggestions in the preparation of revised version of this paper.

References

[1]   | Boersma J.; On a function which is a special case of Meijer’s G-function, Compositio Math., 15 (1962), 34-63.
[2]   | Braaksma B.L.J.; Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compositio Math., 15 (1964), 239-341.
[3]   | Fox C.; The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc., 27(2) (1928), 389-400.
[4]   | Fox C.; The G and H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98 (1961), 395-421.
[5]   | Srivastava H.M. and Manocha H.L.; A Treatise on Generating functions, Halsted Press (Ellis Horwood Ltd., Chichester, U.K.), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
[6]   | Wright E. M.; The asymptotic expansion of the generalized hypergeometric function- I, J. London Math. Soc., 10(4) (1935), 286-293.
[7]   | Wright E.M.; The asymptotic expansion of the generalized hypergeometric function-II, Proc. London Math. Soc.(2), 46 (1940), 389-408.