Very recently Atash and Al-Gonah [1] derived two extension formulas for Lauricella’s function of the second kind of several variables
and
. Now in this research paper we derive two families of transformation formulas for the first kind of Lauricella’s function of several variables
and
with the help of generalized Dixon’s theorem on the sum of the series
obtained earlier by Lavoie
et al. [2]. Some new and known results are also deduced as applications of our main formulas.
Transformation Formulas Lauricella’s function Dixon’s Theorem Kampé de Fériet Function1. Introduction
In 1994, Lavoie et al. [2] , obtained the following generalization of the classical Dixon’s theorem for the series:
,
where denotes the greatest integer less than or equal to x and denotes the usual absolute value of x. The coefficients and are given respectively in [2] . When, (1.1) reduces immediately to the classical Dixon’s theorem [3] , (see also [4] )
We recall that the first kind of the Lauricella hypergeometric function of -variables is defined as [5] :
,
where is the Pochhammer’s symbol defined by [5]
When, (1.3) reduces to the Lauricella function of 2r-variables
.
Clearly, we have, where F2 is Appell’s double hypergeometric function [5]
Next, we recall that the generalized Lauricella function of several variables is defined as [5] :
where
the coefficients, ,;,; for all
are real and positive; abbreviates the array of A parameters; abbreviates
the array of parameters for all with similar interpretations for
and . Note that, when the coefficients in Equation (1.7) equal to 1, the genera-
lized Lauricella function (1.7) reduces to the following multivariable extension of the Kampé de Fériet function [5] :
where
In our present investigation, we shall require the following results [5] :
2. Main Result
In this section, the following transformation formula will be established:
Theorem 2.1. For, the following formula for Lauricella’s function holds true:
where
The coefficients and can be obtained from the tables of Ai,j and Bi,j given in [2] by replacing a and c by and, also the coefficients and can be obtained from the same tables of Ai,j and Bi,j by replacing a and c by and respectively.
Proofs.
In order to prove the Theorem 2.1, let us first prove the following result:
To prove (2.5), denoting the left hand side of (2.5) by I, expanding in a power series as in (1.6) and using the result [5] :
,
we have
.
Now, using the elementary identities [5]
,
we have
.
This completes the proof of (2.5).
Proof of Theorem 2.1. Denoting the left hand side of (2.1) by S, expanding in a power series as in (1.3), adjusting the parameters, using the results (1.11) and (2.5) and by repeating this procedure r-times, we have
where
Now, separating into even and odd powers of by using the elementary identity [5]
,
we have
Finally, if we use the result (1.1), then we obtain the right hand side of the Theorem 2.1. This completes the proof of the Theorem 2.1.
Remark. Taking x = 0 in (2.1), we deduce the following formulas:
Corollary 2.1. For, the following formula for Lauricella’s function holds true:
3. Applications
1) In (2.1) if we take r = 1, then we get a known extension formulas [6] for Lauricella’s function of three variables for
2) In (2.1), if we take, we have
Now, in (3.1) if we use the results (1.12)-(1.16) and simplify, we obtain the following transformation formula:
which for, reduces to
3) Similarly, in (2.6), if we take, we have
which is a generalization of a known result of Bailey [7]
Further, in (3.4) if we take, then we get
4. Conclusion
We conclude our present investigation by remarking that the main results established in this paper can be applied to obtain a large number of transformation formulas for the first kind of Lauricella’s function of several variables. Further, in the formulas (2.1) and (2.6), if we take, then we can obtain two new families of transformation formulas for Lauricella’s functions of several variables
and
for.
Cite this paper
Fadhle B. F. Mohsen,Ahmed Ali Atash,Hussein Saleh Bellehaj, (2016) Transformation Formulas for the First Kind of Lauricella’s Function of Several Variables. Journal of Applied Mathematics and Physics,04,1112-1119. doi: 10.4236/jamp.2016.46115
ReferencesAtash, A.A. and Al-Gonah, A.A. (2016) On Two Extension Formulas for Lauricella’s Function of the Second Kind of Several Variables. Journal of Applied Mathematics and Physics, 4, 571-577. http://dx.doi.org/10.4236/jamp.2016.43062Lavoie, J.L., Grondin, F., Rathie, A.K. and Arora, K. (1994) Generalizations of Dixon’s Theorem on the Sum of a 3F2. Mathematics of Computation, 62, 267-276.Bailey, W.N. (1935) Generalized Hypergeometric Series. Cambridge University Press, Cambridge.Rainville, E.D. (1960) Special Functions. The Macmillan Company, New York.Srivastava, H.M. and Manocha, H.L. (1984) A Treatise on Generating Functions. Halsted Press, New York.Atash A.A. ,et al. (2015)Extension Formulas of Lauricella’s Functions by Applications of Dixon’s Summation Theorem 10, 1007-1018.Bailey, W.N. (1953) On the Sum of Terminating . Quarterly Journal of Mathematics: Oxford Journals, 2, 237-240. http://dx.doi.org/10.1093/qmath/4.1.237