^{1}

^{2, 3}

^{4}

^{1}

^{2}

^{3}

^{4}

Motivated by the extension of classical Gauss's summation theorem for the series

In 1812, Gauss [

The series (

Further, if in (

This series is absolutely convergent for all values of

Gauss hypergeometric function

Also, the Whittaker functions are slightly modified forms of confluent hypergeometric functions. On account of their usefulness, the functions

A natural generalization of

The series (

It should be remarked here that whenever hypergeometric and generalized hypergeometric functions can be summed to be expressed in terms of Gamma functions, the results are very important from a theoretical and an applicable point of view. Only a few summation theorems are available in the literature and it is well known that the classical summation theorems such as of Gauss, Gauss's second, Kummer, and Bailey for the series

As already mentioned that the classical summation theorems such as those of Gauss, Kummer, Gauss's second, and Bailey for the series

In this section, we will mention classical summation theorems for the series

It is not out of place to mention here that Ramanujan independently discovered a great number of the primary classical summation theorems in the theory of hypergeometric series. In particular, he rediscovered well-known summation theorems of Gauss, Kummer, Dougall, Dixon, Saalschütz, and Thomae as well as special cases of the well-known Whipple's transformation. Unfortunately, Ramanujan left us little knowledge as to know how he made his beautiful discoveries about hypergeometric series.

The classical summation theorems mentioned in Section

We now mention here certain very interesting summations by Ramanujan [

For

For

For

For

For

For

For

For

It is easy to see that the series (

The series (

The series (

Also, it can easily be seen that the series (

The series (

Thus by evaluating the hypergeometric series by respective summation theorems, we easily obtain the right hand side of the Ramanujan's summations.

Recently good progress has been done in the direction of generalizing the above-mentioned classical summation theorems (

Notice that, if we denote (

It is very interesting to mention here that, in order to complete the results (

For

On the other hand the following very interesting result for the series

For

Miller [

The aim of this research paper is to establish the extensions of the above mentioned classical summation theorem (

The results are derived with the help of contiguous results of the above mentioned classical summation theorems obtained in a series of three research papers by Lavoie et al. [

The results derived in this paper are simple, interesting, easily established, and may be useful.

The following summation formulas which are special cases of the results (

Contiguous Kummer's theorem [

Contiguous Gauss's Second theorem [

Contiguous Bailey's theorem [

Contiguous Watson's theorem [

Contiguous Dixon's theorem [

Contiguous Whipple's theorem [

In this section, the following extensions of the classical summation theorems will be established. In all these theorems we have

Extension of Kummer's theorem:

Extension of Gauss's second theorem:

Extension of Bailey's theorem:

Extension of Watson's theorem:

First Extension:

Second Extension:

Extension of Dixon's theorem:

Extension of Whipple's theorem:

Extension of (

Extension of (

In order to derive (

Now, it is easy to see that the first and second

In the exactly same manner, the results (

In (

In (

In (

In (

In (

In (

In (

In (

In (

In this section, the following summations, which generalize Ramanujan's summations (

In all the summations, we have

For

For

For

For

For

For

For

For

The series (

The series (

The series (

Also, it can be easily seen that the series (

The series (

Various other applications of these results are under investigations and will be published later.

Further generalizations of the extended summation theorem (

The authors are highly grateful to the referees for carefully reading the manuscript and providing certain very useful suggestions which led to a better presentation of this research article. They are so much appreciated to the College of Science, Sultan Qaboos University, Muscat - Oman for supporting the publication charges of this paper. The first author is supported by the Research Fund of Wonkwang University (2011) and the second author is supported by the research grant (IG/SCI/DOMS/10/03) of Sultan Qaboos University, OMAN.