A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

Abstract

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious series identities.

1. Introduction

Some of the most recent developments are on the use of different techniques for obtaining sums of hypergeometric series. In this paper, we present a new method for calculating the following summations and also a generalized theorem related to these series is investigated. We investigated the following summations formula with some restrictions for the functions :We assume that the function has no poles at , where is an integer ranged from .

As it turns out, the above summation formula for the constant function gives us a new generalized hypergeometric transformation formula of the typeWe further investigated some series closely related to the famous Roger-Ramanujan identities.In our present investigation dealing with the series identity, we shall also make use of other such higher transcendental functions, the Riemann Zeta function and Hurwitz Zeta function, which are defined bywhere is the generalized harmonic numbers defined byIn the above identity we used the notation for denoting the generalized polygamma function of order which is given by the th times logarithmic derivative of Gamma function denotes the famous Euler-Mascheroni constant. The derivatives of generalized harmonic number are given: And we used for the Pochhammer symbol defined (for and in terms of the Gamma function) by The generalized hypergeometric function is defined byThe notation for generalized hypergeometric functions was introduced by Pochhammer in 1870 and modified by Barnes [1] and later by Maier and Slater [2]. A number of notational variations are commonly used. Most common notation is introduced by Graham et al. [3] using square brackets and a semicolon.

The complete Bell polynomials of order are defined as First few terms of these polynomials can be derived by

2. Main Results

Theorem 1. For integers , where and .

Proof. Let us define a function where such thatWe also assume the function has no poles at for each and such thatWe introduce another notation for the th derivative with respect to of the defined function :Furthermore, we see the series also defined for . It can be extended to the interval , because when , where is a positive integer, the further terms of the series just vanish.Making use of certain special properties of Bell polynomials we can evaluate successive derivative of a given function. Let us consider a function which has a Taylor series expansion around ; the detailed procedure of these kinds is extensively discussed in the paper [4]. Following the same process discussed in [5, 6] we can write the following elegant identity by virtue of Bell polynomials: The following identity can be recovered from the same method used in [5, page 8]: represents the th order derivative harmonic number with respect to . The derivative of harmonic numbers can be evaluated by using the formula given in Hence, upon considering (18) and (19), we findThe above identity was derived extensively in [5] and a modified version was calculated in [7]. Now differentiating (13) times with respect to and considering the case , where is a positive integer as well as using the definition from (15), we can deriveMaking use of the identity derived in (20) we findFrom the asymptotic properties of Bell polynomials we haveConsidering the above property in (23), for and are both positive integers, we can readily evaluate the following limits: Considering next few cases, we can move to final relationwhere is defined as in (19). Recalling the properties of Bell polynomials mentioned in (25) if we consider the limit for we observe that for every the limiting value vanishes but it surprisingly gives us limiting value for . Furthermore, expanding the product of generalized harmonic numbers in asymptotic form with Big notation, we can further deduce with and . Substituting the above asymptotic identity in (26) we find After some tedious calculations we findIn view of the known identity for Bell polynomials,We can finally obtain a closed form for the limitSubstituting the limiting value of (26) in (31), we can readily obtain a closed form for the required limitApplying the above limiting value, we haveFinally by combining (22) and (33), we finally obtain Theorem 1.

Theorem 2. For integers ,where and .

Proof. The proof is similar to the previous one. Similarly as before we consider the functionThen by the same process and with same restrictions we can easily obtain Theorem 2.

Theorem 3. For integers , and ,

Proof. Let us take as a constant function in Theorem 1:It proves the first part of Theorem 3. Writing the function and as Hyper geometric functions defined in first section we can deriveFinally combining (37) and (38) we can easily obtain Theorem 3.

Theorem 4. For integers and ,

Proof. Considering Theorem 2 for and using the property of bell polynomials,Taking into account an identity derived in Boyadzhiev’s paper [8] titled “Harmonic Number Identities via Euler’s Transform,” Differentiating the identity in (41) with respect to and using the known formula , Considering the above expression together with (40) we can conclude Theorem 2. Special case of Theorem 2 for can be found in [5]. Theorem 2 also generalizes many identities derived in [9]. The finite summation in the right-hand side of Theorem 2 can be obtained from [7, 10] for different values of . Interested readers can also find some computer assisted proofs of these identities in [11].

Theorem 5. For integers ,

Proof. Setting , and in Theorem 2 and by virtue of Bell polynomialsThe following harmonic number identity can be found in many texts of mathematical literatures. Mainly Chu and De Donno [12] and Paule and Schneider [11] discussed these types of summation formulas in great detail.Differentiating (45) two times with respect to and using the formula involving differentiation of generalized harmonic numbers stated in the first section, Finally substituting the expressions of (45) and (46) in (44), we easily obtain Theorem 5.

Some Special Corollaries

Corollary 6. For integers , Using the following identity (48) given in Volume 2 of Gould’s book [13] and considering Theorem 2 for ; andSome special cases of this formula can be found in the published literature [14–16].

Corollary 7. For integers ,Considering the following identity (50) illustrated in Volume 2 of Gould’s book [13] and using Theorem 2 for ,

Corollary 8. For integers and ,

Proof. The following identity is given in Volume 5 of Gould’s book [13]:Considering Theorem 2 for ; and also using (52) we can immediately find Corollary 8.

Corollary 9. For integers , , and are positive integers,Using the following identity (54) given in Volume 5 of Gould’s book [13] and considering Theorem 2 for Terminating version of these kinds of hypergeometric series goes back to Bailey [15].

Differentiation of Laguerre Polynomials with respect to Its Order. Let be the Laguerre Polynomials defined by

Corollary 10. For integers ,Using the previous identity (55) for Laguerre Polynomials and considering Theorem 2 for we can easily derive Corollary 10.

Corollary 11. For integers , and , Making use of the following identity (58) derived in [17] and considering Theorem 2 for ,

Corollary 12. For integers , and , Making use of the following identity (60) derived in [17] and considering Theorem 2 for ,

Corollary 13. For and integers , where are Stirling numbers of the second kind. Considering the following identity (62) given in [8] for and applying Theorem 2 for, ,

Corollary 14. For and integers ,

Proof. Substituting in Theorem 3,Using Newton’s binomial theorem we haveThis impliesSubstituting the value of from (66) in (64) and after some modifications we can obtain Corollary 14. The modified version of Corollary 14 can be found in [18]. Zeilberger proved quite interesting properties of these types of hypergeometric functions in [19].

Corollary 15. For integers ,

Proof. Substituting and in Theorem 3 and using the property of Bell polynomials,Another classical result, special case of the Vandermonde Convolution, is given byHence we haveConsidering the value of from (70) in (68) we finally recover Corollary 15.

Corollary 16. For integers ,

Proof. Substituting and in Theorem 3 and using the property of Bell polynomials,Considering another classical result illustrated in Gould’s book [13] Volume 5 which first appeared in American Math Monthly, for the case , we have Hence we haveSubstituting the value of from (74) in (72) we can immediately obtain Corollary 16. Various special cases of Corollary 16 can be found in several works of Wimp [20, 21].

Corollary 17. For integers ,

Proof. Substituting and in Theorem 3 and using the property of Bell polynomials,Considering another classical result illustrated in volume 5 of Gould’s book [13] which first appeared in American Math Monthly, considering for the case that hence we have Now differentiating the above identities with respect to and using the definition of polygamma function, We also defineFinally we haveCompiling (76), (77), and (81) we can obtain Corollary 17.

Some Series Identities. -Pochhammer symbol, also called -shifted factorial, is a -analog of the common Pochhammer symbol. It is defined asAnd the binomial coefficients also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients areLet us defineThe operator , used extensively in several references, was recently used by authors in [3, 8, 22].Now by the properties of Bell Polynomials we haveWe can further deduce This implies From the geometric series we have Applying the operator on both sides times,By means of this identity we have Finally Now By virtue of Bell Polynomials we have

Case 1. Substituting in (94), And for is an integer, Finally using the definition of (93) we can derive

Case 2. Substituting in (94), we already derived the expression for . Finally using (94) for we have From previous calculation discussed in (97) we haveBy the same procedure we haveSo finally

Theorems Closely Related to Roger-Ramanujan Identities

Roger-Ramanujan Identities. Rogers Ramanujan identities are given in Roger discovered these identities in 1894 [23, 24], but they were entirely ignored until Ramanujan rediscovered them about 20 years later. A detailed history of these identities can be found in great detail in the survey article written by Andrews [25].

Theorem 18. One has

Proof. From the binomial theorem,Differentiating both sides of (105) with respect to at the point (integer), After some calculations we have Theorem 18:

Theorem 19. One has

Proof. The proof is similar to the previous one. For this case we have to consider the identityDifferentiating (109) with respect to two times when (integer) and considering (99) and (102) together we have,Using the same idea stated above we can calculateBy virtue of Bell polynomials we can derive furtherFinally plugging back the above identity in (110) we can obtain Theorem 19.

3. Conclusion

We would like to enrich the subject by doing further research in this field. In our upcoming paper, we will use the theorems of this paper to prove some series identities related to the Hurwitz Zeta function and also generalize a new theorem. Also we will derive some closed form identities related to double differentiation of Legendre Polynomials with respect to their order and we will derive some new generating function identities involving generalized harmonic numbers and Stirling polynomials.

By using the same techniques derived in this paper one can find various hypergeometric transformation formulas and their -analog. We will consider a curious generalization expression such asin our upcoming paper.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

The author is grateful to Dr. Mourad Ismail for his careful comments and suggestions.