We deduce an explicit representation for the coefficients in a finite expansion of a certain class of generalized hypergeometric functions that contain multiple pairs of numeratorial and denominatorial parameters differing by positive integers. The expansion
alluded to is given in terms of these coefficients and hypergeometric functions of lower order. Applications to Euler and Kummer-type transformations of a subclass of the generalized hypergeometric functions mentioned above together with an extension of the Karlsson-Minton summation formula are provided

Allen Richard Miller

Richard Bruce Paris

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MATHEMATICAL COMMUNICATIONS 205Math. Commun. 17(2012), 205–210On a result related to transformations and summations ofgeneralized hypergeometric seriesAllen Richard Miller1,† and Richard Bruce Paris2,∗1 George Washington University, 1616 18th Street NW, No. 210, Washington, DC20009-2525, USA2 University of Abertay Dundee, Dundee DD1 1HG, UKReceived March 13, 2011; accepted May 28, 2011Abstract. We deduce an explicit representation for the coefficients in a finite expansionof a certain class of generalized hypergeometric functions that contain multiple pairs ofnumeratorial and denominatorial parameters differing by positive integers. The expansionalluded to is given in terms of these coefficients and hypergeometric functions of lower order.Applications to Euler and Kummer-type transformations of a subclass of the generalizedhypergeometric functions mentioned above together with an extension of the Karlsson-Minton summation formula are provided.AMS subject classifications: 33C20Key words: generalized hypergeometric functions and series, transformation and summa-tion formulas1. IntroductionIn [7], the authors have deduced transformation formulas of Euler and Kummer-typerespectively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x),where r pairs of numeratorial and denominatorial parameters differ by positive inte-gers. In addition, in [5, 7], certain quadratic transformations for the former functionas well as a generalization of the Karlsson-Minton summation theorem [3, 10] havebeen derived. All of the transformations mentioned above are extensions of previousresults deduced in [4, 8, 6] and the latter extended summation formula [9] has beenmore efficiently derived in [7] in a simpler form.In the sequel we denote sequences a1, . . . , ap simply by (ap) and define thePochhamer symbol, or shifted factorial, (α)n for complex numbers α and integers n(positive, negative and zero) by(α)n ≡ Γ(α + n)Γ(α) ,where Γ(α) is the gamma function. Furthermore, we define products of Pochhammersymbols by((ap))n ≡ (a1)n . . . (ap)n,†Passed away on 15 August 2010.∗Corresponding author. Email addresses: r.paris@abertay.ac.uk (R.B. Paris)http://www.mathos.hr/mc c©2012 Department of Mathematics, University of Osijek206 A.R.Miller and R.B. Pariswhere when p = 0 the product is empty and reduces to unity. We shall adopt thenotation {nk} for the Stirling numbers of the second kind as employed by Graham etal. [2, Section 6]. Recall that Stirling numbers of the second kind {nk} represent thenumber of ways to partition n objects into k nonempty subsets. Thus {00} ≡ 1 and{n0} = 0 when integer n > 0.With this notation all of the results alluded to above are consequences of thefollowing theorem whose proof is found in [7, Lemma 4]. This theorem enablesan r+sFr+1(x) hypergeometric function, where in the sequel s = 1, 2 and r pairsof numeratorial and denominatorial parameters differ by positive integers, to beexpressed as a finite sum of sF1(x) functions.Theorem 1. For a nonnegative integer s let (as) denote a parameter sequence con-taining s elements, where when s = 0 the sequence is empty. Let (as + k) denotethe sequence when k is added to each element of (as). Let F(x) denote the gener-alized hypergeometric function with r numeratorial and denominatorial parametersdiffering by positive integers (mr), namelyF(x) ≡ r+sFr+1((as),c,(fr + mr)(fr)∣∣∣∣ x),where convergence of the series representation for the latter occurs in an appropriatedomain depending on the values of s and the elements of the parameter sequence (as).ThenF(x) = 1A0m∑k=0xkAk((as))k(c)ksF1((as + k)c + k∣∣∣∣ x),where m = m1 + · · ·+ mr, the coefficients Ak are defined byAk ≡m∑j=k{jk}σm−j , (1)and the σj (0 ≤ j ≤ m) are generated by the relation(f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj . (2)Although it is evident thatA0 = (f1)m1 · · · (fr)mr , Am = 1, (3)the coefficients Ak for 0 < k < m are otherwise defined implicitly by (1) and (2).It is the purpose of this brief communication to obtain in Section 2 an explicitrepresentation (Theorem 2) for the coefficients Ak (0 ≤ k ≤ m). Then in Section 3we shall record a few of the salient results that are consequences of Theorems 1 and2.A result related to the hypergeometric series 2072. The coefficients AkWe prove the following.Theorem 2. Suppose (fr) is a nonempty sequence of complex numbers and (mr) asequence of positive integers such that m ≡ m1 + · · ·+ mr. Suppose further thatAk ≡m∑j=k{jk}σm−j (0 ≤ k ≤ m),where the σj (0 ≤ j ≤ m) are generated by the relation(f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj .ThenAk =(−1)kA0k! r+1Fr( −k, (fr + mr)(fr)∣∣∣∣ 1), (4)where 0 ≤ k ≤ m and A0 is given by (3).Proof. We define the monic polynomial P (x) of degree m byP (x) ≡ (f1 + x)m1 . . . (fr + x)mr = σm + σm−1x + · · ·+ σ1xm−1 + σ0xm,where σ0 = 1. It then follows that σm−j = P (j)(0)/j! (0 ≤ j ≤ m) and consequentlyAk =m∑j=k{jk}P (j)(0)j!.From [1, 24.1.1 (C)] and [1, 24.1.4 (C)], we have∆kP (x) =k∑j=0(−1)k−j(kj)P (x + j) = k!m∑j=k{jk}P (j)(x)j!,where ∆ denotes the forward difference operator used in numerical analysis. HenceAk =∆kP (0)k!=1k!k∑j=0(−1)k−j(kj)P (j).Now sinceP (j) = (f1 + j)m1 . . . (fr + j)mrand (α + j)p = (α)p (α + p)j/(α)j for positive integers p, we obtainAk =(−1)kk!(f1)m1 . . . (fr)mrk∑j=0(−1)j(kj)(f1 + m1)j(f1)j. . .(fr + mr)j(fr)j,orAkA0=(−1)kk!k∑j=0(−k)jj!((fr + mr))j((fr))jwhich evidently completes the proof.208 A.R.Miller and R.B. ParisCorollary 1. Let (fr) and (mr) be sequences as in Theorem 2. Thenr+1Fr(−m, (fr + mr)(fr)∣∣∣∣ 1)=(−1)m m!(f1)m1 . . . (fr)mr, (5)where m = m1 + · · ·+ mr.Proof. In (4) set k = m and then use (3).We remark that Karlsson [3] proved (5) by employing other methods.3. Transformation and summation formulasTheorem 2 allows us to state in a somewhat simplified and more elegant form severalresults obtained in [7] by use of Theorem 1. Thus, for example, we have the followingtheorem that provides transformation formulas of Euler and Kummer-type respec-tively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x), inwhich r pairs of numeratorial and denominatorial parameters differ by positive in-tegers.Theorem 3. Let (mr) be a nonempty sequence of positive integers such that m =m1 + · · ·+ mr. Then if b 6= fj (1 ≤ j ≤ r), (λ)m 6= 0, where λ ≡ c− b−m, we havethe transformation formulasr+2Fr+1(a, b,c,(fr + mr)(fr)∣∣∣∣ x)= (1− x)−am+2Fm+1(a, λ,c,(ξm + 1)(ξm)∣∣∣∣xx− 1),where |x| < 1, Rex < 12 , andr+1Fr+1(b,c,(fr + mr)(fr)∣∣∣∣ x)= exm+1Fm+1(λ,c,(ξm + 1)(ξm)∣∣∣∣− x),where |x| < ∞. The (ξm) are the nonvanishing zeros of the associated parametricpolynomial Qm(t) of degree m defined byQm(t) ≡m∑k=0Ak(b)k(t)k(λ− t)m−k,where the Ak (0 ≤ k ≤ m) are given by (4).The Karlsson-Minton summation formula [3, 10], which is also a consequence ofTheorem 1 as shown in [7], states that for Re (−a) > m− 1r+1Fr(a, b,b + 1,(fr + mr)(fr)∣∣∣∣ 1)=Γ(1 + b)Γ(1− a)Γ(1 + b− a)(f1 − b)m1 . . . (fr − b)mr(f1)m1 . . . (fr)mr. (6)An elegant extension of this summation theorem may be obtained directly fromTheorem 1 by setting x = 1 and s = 2 in the latter, employing the Gauss summationtheorem for 2F1(1) and utilization of Theorem 2. Thus we obtain the following.A result related to the hypergeometric series 209Theorem 4. Suppose (mr) is a sequence of positive integers such that m = m1 +· · ·+ mr. Then, provided that Re (c− a− b) > m we haver+2Fr+1(a, b,c,(fr + mr)(fr)∣∣∣∣ 1)=Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)m∑k=0r+1Fr(−k, (fr + mr)(fr)∣∣∣∣ 1)(a)k(b)k(1 + a + b− c)k k! . (7)If in (7) we set c = b, the right-hand side of the latter vanishes and we have thefollowing.Corollary 2. Suppose (mr) is a sequence of positive integers such that m = m1 +· · ·+ mr. Thenr+1Fr(a, (fr + mr)(fr)∣∣∣∣ 1)= 0, Re (−a) > m. (8)This last result was also obtained by Karlsson [3] who used other methods.If c = b + 1, then (7) reduces tor+1Fr(a, b,b + 1,(fr + mr)(fr)∣∣∣∣ 1)=Γ(1 + b)Γ(1− a)Γ(1 + b− a)m∑k=0r+1Fr(−k, (fr + mr)(fr)∣∣∣∣ 1)(b)kk!,where Re (−a) > m− 1. Then upon employing (6) and (8) we obtain the following.Corollary 3. Suppose (mr) is a sequence of positive integers. Then∞∑k=0r+1Fr(−k, (fr + mr)(fr)∣∣∣∣ 1)(b)kk!=(f1 − b)m1 . . . (fr − b)mr(f1)m1 . . . (fr)mr.When fj = f , mj = m (1 ≤ j ≤ r), we find∞∑k=0r+1Fr(−k, f + m,f,. . . ,. . . ,f + mf∣∣∣∣ 1)(b)kk!=((f − b)m(f)m)r,which is a generalization of the Vandermonde-Chu convolution theorem; see [11,pp. 30–31].References[1] M.Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, NewYork, 1965.[2] R.L.Graham, D.E.Knuth, O.Patashnik, Concrete Mathematics, Addison-Wesley, Upper Saddle River, 1994.[3] P.W.Karlsson, Hypergeometric functions with integral parameter differences, J.Math. Phys. 12(1971), 270–271.[4] A.R.Miller, Certain summation and transformation formulas for generalized hyper-geometric series, J. Comput. Appl. Math. 231(2009), 964–972.210 A.R.Miller and R.B. Paris[5] A.R.Miller, R.B. Paris, Certain transformations and summations for generalizedhypergeometric series with integral parameter differences, Int. Trans. Spec. Funct.22(2011), 67–77.[6] A.R.Miller, R.B. Paris, Euler-type transformations for the generalized hypergeo-metric function r+2Fr+1(x), Z. Angew. Math. Phys. 62(2011), 31–45.[7] A.R.Miller, R.B. Paris, Transformation formulas for the generalized hypergeomet-ric function with integral parameter differences, Rocky Mountain J. Math., to appear.[8] A.R.Miller, R.B. Paris, A generalized Kummer-type transformation for the pFp(x)hypergeometric function, Canadian Math. Bulletin, to appear.[9] A.R.Miller, H.M. Srivastava, Karlsson-Minton summation theorems for the gen-eralized hypergeometric series of unit argument, Int. Trans. Spec. Funct. 21(2010),603–612.[10] B.M.Minton, Generalized hypergeometric function of unit argument, J. Math. Phys.11(1970), 1375–1376.[11] H.M. Srivastava, H. L.Manocha, A Treatise on Generating Functions, Ellis Hor-wood, Chichester, 1984.

English

On a result related to transformations and summations of generalized hypergeometric series

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MATHEMATICAL COMMUNICATIONS 205Math. Commun. 17(2012), 205–210On a result related to transformations and summations ofgeneralized hypergeometric seriesAllen Richard Miller1,† and Richard Bruce Paris2,∗1 George Washington University, 1616 18th Street NW, No. 210, Washington, DC20009-2525, USA2 University of Abertay Dundee, Dundee DD1 1HG, UKReceived March 13, 2011; accepted May 28, 2011Abstract. We deduce an explicit representation for the coefficients in a finite expansionof a certain class of generalized hypergeometric functions that contain multiple pairs ofnumeratorial and denominatorial parameters differing by positive integers. The expansionalluded to is given in terms of these coefficients and hypergeometric functions of lower order.Applications to Euler and Kummer-type transformations of a subclass of the generalizedhypergeometric functions mentioned above together with an extension of the Karlsson-Minton summation formula are provided.AMS subject classifications: 33C20Key words: generalized hypergeometric functions and series, transformation and summa-tion formulas1. IntroductionIn [7], the authors have deduced transformation formulas of Euler and Kummer-typerespectively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x),where r pairs of numeratorial and denominatorial parameters differ by positive inte-gers. In addition, in [5, 7], certain quadratic transformations for the former functionas well as a generalization of the Karlsson-Minton summation theorem [3, 10] havebeen derived. All of the transformations mentioned above are extensions of previousresults deduced in [4, 8, 6] and the latter extended summation formula [9] has beenmore efficiently derived in [7] in a simpler form.In the sequel we denote sequences a1, . . . , ap simply by (ap) and define thePochhamer symbol, or shifted factorial, (α)n for complex numbers α and integers n(positive, negative and zero) by(α)n ≡ Γ(α+ n)Γ(α) ,where Γ(α) is the gamma function. Furthermore, we define products of Pochhammersymbols by((ap))n ≡ (a1)n . . . (ap)n,†Passed away on 15 August 2010.∗Corresponding author. Email addresses: r.paris@abertay.ac.uk (R.B. Paris)http://www.mathos.hr/mc c©2012 Department of Mathematics, University of Osijek206 A.R.Miller and R.B. Pariswhere when p = 0 the product is empty and reduces to unity. We shall adopt thenotation {nk} for the Stirling numbers of the second kind as employed by Graham etal. [2, Section 6]. Recall that Stirling numbers of the second kind {nk} represent thenumber of ways to partition n objects into k nonempty subsets. Thus {00} ≡ 1 and{n0} = 0 when integer n > 0.With this notation all of the results alluded to above are consequences of thefollowing theorem whose proof is found in [7, Lemma 4]. This theorem enablesan r+sFr+1(x) hypergeometric function, where in the sequel s = 1, 2 and r pairsof numeratorial and denominatorial parameters differ by positive integers, to beexpressed as a finite sum of sF1(x) functions.Theorem 1. For a nonnegative integer s let (as) denote a parameter sequence con-taining s elements, where when s = 0 the sequence is empty. Let (as + k) denotethe sequence when k is added to each element of (as). Let F(x) denote the gener-alized hypergeometric function with r numeratorial and denominatorial parametersdiffering by positive integers (mr), namelyF(x) ≡ r+sFr+1((as),c,(fr +mr)(fr)∣∣∣∣x) ,where convergence of the series representation for the latter occurs in an appropriatedomain depending on the values of s and the elements of the parameter sequence (as).ThenF(x) = 1A0m∑k=0xkAk((as))k(c)ksF1((as + k)c+ k∣∣∣∣x) ,where m = m1 + · · ·+mr, the coefficients Ak are defined byAk ≡m∑j=k{jk}σm−j , (1)and the σj (0 ≤ j ≤ m) are generated by the relation(f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj . (2)Although it is evident thatA0 = (f1)m1 · · · (fr)mr , Am = 1, (3)the coefficients Ak for 0 < k < m are otherwise defined implicitly by (1) and (2).It is the purpose of this brief communication to obtain in Section 2 an explicitrepresentation (Theorem 2) for the coefficients Ak (0 ≤ k ≤ m). Then in Section 3we shall record a few of the salient results that are consequences of Theorems 1 and2.A result related to the hypergeometric series 2072. The coefficients AkWe prove the following.Theorem 2. Suppose (fr) is a nonempty sequence of complex numbers and (mr) asequence of positive integers such that m ≡ m1 + · · ·+mr. Suppose further thatAk ≡m∑j=k{jk}σm−j (0 ≤ k ≤ m),where the σj (0 ≤ j ≤ m) are generated by the relation(f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj .ThenAk =(−1)kA0k! r+1Fr( −k, (fr +mr)(fr)∣∣∣∣ 1) , (4)where 0 ≤ k ≤ m and A0 is given by (3).Proof. We define the monic polynomial P (x) of degree m byP (x) ≡ (f1 + x)m1 . . . (fr + x)mr = σm + σm−1x+ · · ·+ σ1xm−1 + σ0xm,where σ0 = 1. It then follows that σm−j = P (j)(0)/j! (0 ≤ j ≤ m) and consequentlyAk =m∑j=k{jk}P (j)(0)j!.From [1, 24.1.1 (C)] and [1, 24.1.4 (C)], we have∆kP (x) =k∑j=0(−1)k−j(kj)P (x+ j) = k!m∑j=k{jk}P (j)(x)j!,where ∆ denotes the forward difference operator used in numerical analysis. HenceAk =∆kP (0)k!=1k!k∑j=0(−1)k−j(kj)P (j).Now sinceP (j) = (f1 + j)m1 . . . (fr + j)mrand (α+ j)p = (α)p (α+ p)j/(α)j for positive integers p, we obtainAk =(−1)kk!(f1)m1 . . . (fr)mrk∑j=0(−1)j(kj)(f1 +m1)j(f1)j. . .(fr +mr)j(fr)j,orAkA0=(−1)kk!k∑j=0(−k)jj!((fr +mr))j((fr))jwhich evidently completes the proof.208 A.R.Miller and R.B. ParisCorollary 1. Let (fr) and (mr) be sequences as in Theorem 2. Thenr+1Fr(−m, (fr +mr)(fr)∣∣∣∣ 1) = (−1)mm!(f1)m1 . . . (fr)mr , (5)where m = m1 + · · ·+mr.Proof. In (4) set k = m and then use (3).We remark that Karlsson [3] proved (5) by employing other methods.3. Transformation and summation formulasTheorem 2 allows us to state in a somewhat simplified and more elegant form severalresults obtained in [7] by use of Theorem 1. Thus, for example, we have the followingtheorem that provides transformation formulas of Euler and Kummer-type respec-tively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x), inwhich r pairs of numeratorial and denominatorial parameters differ by positive in-tegers.Theorem 3. Let (mr) be a nonempty sequence of positive integers such that m =m1 + · · ·+mr. Then if b 6= fj (1 ≤ j ≤ r), (λ)m 6= 0, where λ ≡ c− b−m, we havethe transformation formulasr+2Fr+1(a, b,c,(fr +mr)(fr)∣∣∣∣x) = (1− x)−am+2Fm+1(a, λ,c, (ξm + 1)(ξm)∣∣∣∣ xx− 1),where |x| < 1, Rex < 12 , andr+1Fr+1(b,c,(fr +mr)(fr)∣∣∣∣x) = exm+1Fm+1(λ,c,(ξm + 1)(ξm)∣∣∣∣− x) ,where |x| < ∞. The (ξm) are the nonvanishing zeros of the associated parametricpolynomial Qm(t) of degree m defined byQm(t) ≡m∑k=0Ak(b)k(t)k(λ− t)m−k,where the Ak (0 ≤ k ≤ m) are given by (4).The Karlsson-Minton summation formula [3, 10], which is also a consequence ofTheorem 1 as shown in [7], states that for Re (−a) > m− 1r+1Fr(a, b,b+ 1,(fr +mr)(fr)∣∣∣∣ 1) = Γ(1 + b)Γ(1− a)Γ(1 + b− a) (f1 − b)m1 . . . (fr − b)mr(f1)m1 . . . (fr)mr . (6)An elegant extension of this summation theorem may be obtained directly fromTheorem 1 by setting x = 1 and s = 2 in the latter, employing the Gauss summationtheorem for 2F1(1) and utilization of Theorem 2. Thus we obtain the following.A result related to the hypergeometric series 209Theorem 4. Suppose (mr) is a sequence of positive integers such that m = m1 +· · ·+mr. Then, provided that Re (c− a− b) > m we haver+2Fr+1(a, b,c,(fr +mr)(fr)∣∣∣∣ 1)=Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)m∑k=0r+1Fr(−k, (fr +mr)(fr)∣∣∣∣ 1) (a)k(b)k(1 + a+ b− c)k k! . (7)If in (7) we set c = b, the right-hand side of the latter vanishes and we have thefollowing.Corollary 2. Suppose (mr) is a sequence of positive integers such that m = m1 +· · ·+mr. Thenr+1Fr(a, (fr +mr)(fr)∣∣∣∣ 1) = 0, Re (−a) > m. (8)This last result was also obtained by Karlsson [3] who used other methods.If c = b+ 1, then (7) reduces tor+1Fr(a, b,b+ 1,(fr +mr)(fr)∣∣∣∣ 1) = Γ(1 + b)Γ(1− a)Γ(1 + b− a)m∑k=0r+1Fr(−k, (fr +mr)(fr)∣∣∣∣ 1) (b)kk! ,where Re (−a) > m− 1. Then upon employing (6) and (8) we obtain the following.Corollary 3. Suppose (mr) is a sequence of positive integers. Then∞∑k=0r+1Fr(−k, (fr +mr)(fr)∣∣∣∣ 1) (b)kk! = (f1 − b)m1 . . . (fr − b)mr(f1)m1 . . . (fr)mr .When fj = f , mj = m (1 ≤ j ≤ r), we find∞∑k=0r+1Fr(−k, f +m,f,. . . ,. . . ,f +mf∣∣∣∣ 1) (b)kk! =((f − b)m(f)m)r,which is a generalization of the Vandermonde-Chu convolution theorem; see [11,pp. 30–31].References[1] M.Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, NewYork, 1965.[2] R.L.Graham, D.E.Knuth, O.Patashnik, Concrete Mathematics, Addison-Wesley, Upper Saddle River, 1994.[3] P.W.Karlsson, Hypergeometric functions with integral parameter differences, J.Math. Phys. 12(1971), 270–271.[4] A.R.Miller, Certain summation and transformation formulas for generalized hyper-geometric series, J. Comput. Appl. Math. 231(2009), 964–972.210 A.R.Miller and R.B. Paris[5] A.R.Miller, R.B. Paris, Certain transformations and summations for generalizedhypergeometric series with integral parameter differences, Int. Trans. Spec. Funct.22(2011), 67–77.[6] A.R.Miller, R.B. Paris, Euler-type transformations for the generalized hypergeo-metric function r+2Fr+1(x), Z. Angew. Math. Phys. 62(2011), 31–45.[7] A.R.Miller, R.B. Paris, Transformation formulas for the generalized hypergeomet-ric function with integral parameter differences, Rocky Mountain J. Math., to appear.[8] A.R.Miller, R.B. Paris, A generalized Kummer-type transformation for the pFp(x)hypergeometric function, Canadian Math. Bulletin, to appear.[9] A.R.Miller, H.M. Srivastava, Karlsson-Minton summation theorems for the gen-eralized hypergeometric series of unit argument, Int. Trans. Spec. Funct. 21(2010),603–612.[10] B.M.Minton, Generalized hypergeometric function of unit argument, J. Math. Phys.11(1970), 1375–1376.[11] H.M. Srivastava, H. L.Manocha, A Treatise on Generating Functions, Ellis Hor-wood, Chichester, 1984.

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