Kim, Yong S.

Pandey, U.

Paris, Richard B.

Rathie, Arjun K.

Abertay Research Portal

Hacettepe Journal of Mathematics and StatisticsVolume 42 (3) (2013), 281 – 287SOME SUMMATION FORMULASFOR THE HYPERGEOMETRICSERIES r+2Fr+1(12)Y. S. Kim∗, A. K. Rathie†, U. Pandey‡ and R. B. Paris§ ¶Received 16 : 06 : 2011 : Accepted 22 : 06 : 2012AbstractThe aim of this paper is to obtain explicit expressions of the generalizedhypergeometric functionr+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]for j = 0,±1, . . . ,±5, where r pairs of numeratorial and denominatorialparameters differ by positive integers mr. The results are derived withthe help of an expansion in terms of a finite sum of 2F1(12) functions anda generalization of Gauss’ second summation theorem due to Lavoie etal. [J. Comput. Appl. Math. 72, 293–300 (1996)]. Some special andlimiting cases are also given.Keywords: Generalized hypergeometric series, Generalized Gauss summation theorem2000 AMS Classification: Primary 33C20, Secondary 33C15∗Department of Mathematics Education, Wonkwang University, Iksan, Korea.E-Mail: yspkim@wonkwang.ac.kr†Department of Mathematics, Central University of Kerala, Kasaragad 671328, Kerala, India.E-Mail: akrathie@rediffmail.com‡Department of Mathematics, Marudhar Engineering College, Raisar, Bikaner, RajasthanState, India. E-mail: pandey.up@rediffmail.com§University of Abertay Dundee, Dundee DD1 1HG, UK. E-Mail: r.paris@abertay.ac.uk¶Corresponding author282 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Paris1. IntroductionThe generalized hypergeometric function with p numeratorial and q denominatorialparameters is defined by the series [16, p. 41](1.1) pFq[a1, a2, . . . , apb1, b2, . . . , bq;x]=∞∑n=0(a1)n(a2)n . . . (ap)n(b1)n(b2)n . . . (bq)nxnn!,where (a)n = Γ(a+ n)/Γ(a) is the Pochhammer symbol (or ascending factorial). Whenq = p this series converges for |x| < ∞, but when q = p − 1 convergence occurs when|x| < 1 (unless the series terminates). In what follows we shall adopt the convention ofwriting the finite sequence of parameters (a1, . . . , ap) simply by (ap).It is well known that whenever hypergeometric functions reduce to gamma functions,the results are very important from the applications point of view. Thus, the classicaltheorems of Gauss, Kummer and Bailey for the series 2F1, and of Watson, Dixon, Whip-ple and Saalschütz for the series 3F2, and others, play an important role in the theory ofhypergeometric and generalized hypergeometric series. In [3, 4, 5], Lavoie et al. consid-ered generalizations of some of the above-mentioned classical summations. In particular,they obtained a generalization of Gauss’ second summation theorem, given by2F1[a, b12a+ 12b+ 12; 12]= π12Γ( 12a+ 12b+ 12)Γ( 12a+ 12)Γ( 12b+ 12)provided 12a+ 12b+ 126= 0,−1,−2, . . . , in the following form1.1. Theorem. [5] Provided 12(a+ b+ j + 1) 6= 0,−1,−2, . . ., we have the summation(1.2) 2F1[a, b12(a+ b+ j + 1); 12]= π12Γ( 12a+ 12b+ 12j + 12)Γ( 12a− 12b− 12j + 12)Γ( 12a− 12b+ 12+ 12|j|)×{Aj(a, b)Γ( 12a+ 12)Γ( 12b+ δj+1)+Bj(a, b)Γ( 12a)Γ( 12b+ δj)}for integer j. As usual, [x] denotes the greatest integer less than or equal to x, its modulusis denoted by |x| and we have defined(1.3) δj ≡ 12 j − [12j].The coefficients Aj(a, b) and Bj(a, b) are displayed in Table 1 for 0 ≤ j ≤ 5, where thecoefficients for j < 0 satisfy(1.4) A−j(a, b) = (−1)jAj(a, b), B−j(a, b) = (−1)j+1Bj(a, b).Table 1. The coefficients Aj and Bj for 0 ≤ j ≤ 5.j Aj(a, b) Bj(a, b)0 1 01 −1 12 12(a+ b− 1) −23 − 12(3a+ b− 2) 12(3b+ a− 2)4 14(a+ b− 1)(a+ b− 3) + ab −2(a+ b− 1)5 − 14(a+ b− 2)(5a+ b− 4) + a(b+ 1) 14(a+ b− 2)(5b+ a− 4) + b(a+ 1)Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 283Recently Miller [7], Miller and Paris [8, 9] and Miller and Srivastava [12] studiedthe generalized hypergeometric series (including integer parameter differences) and ob-tained numerous transformation and summation formulas for the series r+2Fr+1(z) andr+1Fr+1(z). In our present investigation, we shall require the following result establishedin [8] expressing r+2Fr+1(z), with r numeratorial and denominatorial parameters dif-fering by the positive integers (mr), in terms of a finite sum of Gauss hypergeometricfunctions.1.2. Theorem. [8] Let (mr) denote a set of positive integers with m = m1 + · · ·+mr.Then, when |z| < 1, we have(1.5) r+2Fr+1[a, b,c,(fr +mr)(fr); z]=m∑k=0(a)k(b)k(c)kzkCk(r) 2F1[a+ k, b+ kc+ k; z],where the coefficients Ck(r) are defined by [10](1.6) Ck(r) =(−1)kk!r+1Fr[−k, (fr +mr)(fr); 1].The expansion also holds when z = 1 provided Re (c− a− b) > m.Alternatively, the coefficients can be expressed in the form [8](1.7) Ck(r) ≡1Λm∑j=k{jk}σm−j , Λ = (f1)m1 . . . (fr)mr ,with C0(r) = 1, Cm(r) = 1/Λ, where {jk} denotes the Stirling number of the second kindand the σj (0 ≤ j ≤ m) are generated by the relation(1.8) (f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj .The result (1.5) and (1.6) can also be deduced as a particular case of the more generalexpansion given by Luke in [6, Eq. (5.10.2(4))] combined with the fact that Ck(r) = 0for k > m [2, 10].1.3. Remark. If we set z = 1 in (1.5), we immediately obtain the generalization of theKarlsson-Minton summation theorem [11, 12]r+2Fr+1[a, b,c,(fr +mr)(fr); 1]=Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)m∑k=0(−)k(a)k(b)kCk(r)(1 + a+ b− c)kprovided Re (c−a−b) > m. If c = b+1 and use is made of the alternative representationof the coefficients Ck(r) in (1.7), this reduces to the Karlsson-Minton summation formulagiven by [11, 13]r+2Fr+1[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)mrwhen Re (−a) > m− 1.Our aim in this paper is to obtain explicit expressions ofr+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]for complex parameters a, b and (fr) and for j = 0,±1, . . . ,±5, where r pairs of numera-torial and denominatorial parameters differ by positive integers (mr). For this purpose weshall make use of the expansion (1.5) combined with the generalization in (1.2) of Gauss’284 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Parissecond summation theorem. Several special and limiting cases of our main findings arealso presented.2. Summation formulasOur main result is given by the following theorem.2.1. Theorem. Let (mr) be a set of positive integers with m = m1 + · · · + mr and letj be an integer. Then, the generalized hypergeometric function of argument 12and withr pairs of numeratorial and denominatorial parameters differing by positive integers hasthe summation(2.1) r+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]= π12Γ( 12a+ 12b+ 12j + 12)Γ( 12a− 12b− 12j + 12)Γ( 12a− 12b+ 12|j|+ 12)m∑k=0(a)k(b)kCk(r)(12)k×{Aj(a+ k, b+ k)Γ( 12a+ 12k + 12)Γ( 12b+ 12k + δj+1)+Bj(a+ k, b+ k)Γ( 12a+ 12k)Γ( 12b+ 12k + δj)},where the coefficients Aj and Bj for −5 ≤ j ≤ 5 can be obtained from the Aj(a, b) andBj(a, b) in Table 1 and use of (1.4) simply by putting a 7→ a + k and b 7→ b + k, and δjis defined in (1.3).Proof. If we set z = 12and c = 12(a+ b+ j + 1), where j is an integer, in (1.5) we obtainr+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]=m∑k=0(a)k(b)kCk(r)2k( 12a+ 12b+ 12j + 12)k2F1[a+ k, b+ k12a+ 12b+ 12j + 12+ k; 12].Application of (1.2) then leads to the result stated in (2.1). 2 To simplify the presentation of specific cases, we define the quantities D(1)± (k) andD(2)± (k) byD(1)± (k) ≡1a± b(Γ( 12a+ 12+ 12k)Γ( 12b+ 12k)± Γ( 12a+ 12k)Γ( 12b+ 12+ 12k)),D(2)± (k) ≡1(a± b)2 − 1((a+b+2k−1)Γ( 12a+ 12k)Γ( 12b+ 12k)± 4Γ( 12a+ 12+ 12k)Γ( 12b+ 12+ 12k)).Then, upon use of the duplication formula for the gamma function(2.2)√π Γ(2z) = 22z−1Γ(z)Γ(z + 12),we have the following summations.2.2. Corollary. The summations in (2.1) for j = 0,±1,±2, respectively, take the form(2.3)r+2Fr+1[a, b,12(a+b+1),(fr +mr)(fr); 12]=2a+b−2√πΓ( 12a+ 12b+ 12)Γ(a)Γ(b)m∑k=02kCk(r) Γ(12a+ 12k)Γ( 12b+ 12k),Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 285(2.4)r+2Fr+1[a, b,12(a+b+1)± 12,(fr +mr)(fr); 12]=2a+b−1√πΓ( 12a+ 12b+1)Γ(a)Γ(b)m∑k=02kCk(r)D(1)∓ (k),(2.5)r+2Fr+1[a, b,12(a+b+1)± 1,(fr +mr)(fr); 12]=2a+b−1√πΓ( 12a+ 12b+ 32)Γ(a)Γ(b)m∑k=02kCk(r)D(2)∓ (k).When r = 0, the sequences (fr) and (fr +mr) are empty so that m = 0 and (2.3)–(2.5)reduce to (1.2).2.3. Remark. The results for j = −2, −1, 0 and 1 have been obtained by Miller andParis [8]. However, the results for j = −2 and j = −1 are given here in corrected forms.3. Specific examplesAs specific examples, we let r = 1, m1 = 1 and f1 = f , so that C0(1) = 1, C1(1) = 1/f ,and thus obtain(3.1) 3F2[a, b,12(a+ b+ 1)f + 1f; 12]= π12 Γ( 12a+ 12b+ 12){1Γ( 12a+ 12)Γ( 12b+ 12)+2/fΓ( 12a)Γ( 12b)},(3.2) 3F2[a, b,12(a+ b+ 1)± 12,f + 1f; 12]= 2π12Γ( 12a+ 12b+ 1)a∓ b{(a∓ f)/fΓ( 12a+ 12)Γ( 12b)∓ (b∓ f)/fΓ( 12a)Γ( 12b+ 12)},(3.3) 3F2[a, b,12(a+ b+ 1)± 1,f + 1f; 12]= 2π12Γ( 12a+ 12b+ 32)(a∓ b)2 − 1{a+ b− 1∓ (2ab)/fΓ( 12a+ 12)Γ( 12b+ 12)∓ 4∓ 2(a+ b+ 1)/fΓ( 12a)Γ( 12b)}.3.1. Remark. The summation in (3.2) with the upper signs has been obtained previouslyin [10] and [15].The summations in (2.1) corresponding to j ≥ 1 are not valid when a−b = 0,±1, . . . ,±( 12j − 12) for odd j, and when a − b = ± 12, . . . ,±( 12j − 12) for even j. In such cases wemay use l’Hôpital’s rule and the duplication formula (2.2) to obtain the limiting casesgiven below.286 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Paris3.2. Corollary. When a = b, for example, we find the following summations correspond-ing to j = 1, 3 and 5 respectively:(3.4) r+2Fr+1[a, a,a+ 1,(fr +mr)(fr); 12](3.5) = 2a−1am∑k=0Ck(r)(a)k{ψ( 12a+ 12k + 12)− ψ( 12a+ 12k)},(3.6) r+2Fr+1[a, a,a+ 2,(fr +mr)(fr); 12]= 2a−1a(a+ 1)m∑k=0Ck(r)(a)k{(2a + 2k − 1){ψ( 12a + 12k)− ψ( 12a + 12k + 12)}+ 2}andr+2Fr+1[a, a,a+ 3,(fr +mr)(fr); 12]= 2a−1a(a+ 1)(a+ 2)m∑k=0Ck(r)(a)k{{(a+ k − 12)2 + 14}{ψ( 12a+ 12k + 12)− ψ( 12a+ 12k)} − (a+ k − 12)},where ψ is the digamma or psi function.3.3. Corollary. Similarly, if a− b = 1 when j = 2 and 4 we find the summations:(3.7) r+2Fr+1[a, a+ 1,a+ 2,(fr +mr)(fr); 12]= 2aa(a+ 1)m∑k=0Ck(r)(a+ 1)k{ψ( 12a+ 12k + 12)− ψ( 12a+ 12k)− 1a+ k}and(3.8) r+2Fr+1[a, a+ 1,a+ 3,(fr +mr)(fr); 12]= 2aa(a+ 1)(a+ 2)m∑k=0Ck(r)(a+ 1)k{(a+ k){ψ( 12a+ 12k)− ψ( 12a+ 12k + 12)}+ 1 + 12(a+ k)}.Finally, it is worthy of mention that when r = 0 we retrieve respectively from (3.4)–(3.8), or from (1.2), the following results for the series 2F1(12):2F1[a, aa+ 1; 12]= 2a−1a{ψ( 12a+ 12)− ψ( 12a)},2F1[a, aa+ 2; 12]= 2a−1a(a+ 1){(2a− 1){ψ( 12a)− ψ( 12a+ 12)}+ 2},2F1[a, aa+ 3; 12]= 2a−1a(a+ 1)(a+ 2){{(a− 12)2 + 14}{ψ( 12a+ 12)− ψ( 12a)} − (a− 12)},2F1[a, a+ 1a+ 2; 12]= 2aa(a+ 1){ψ( 12a+ 12)− ψ( 12a)− 1a}Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 287and2F1[a, a+ 1a+ 3; 12]= 2aa(a+ 1)(a+ 2){a{ψ( 12a)− ψ( 12a+ 12)}+ 1 + 12a}.3.4. Remark. The first of the above 2F1(12) summations is found in [1, p. 557] and in[14, p. 492] written in slightly different form, but the remaining results are believed tobe new.AcknowledgmentsOne of the authors (YSK) acknowledges the support of the Wonkwang UniversityResearch Fund (2012).References[1] Abramowitz, M. and Stegun, I. A. (Eds.) Handbook of Mathematical Functions (Dover, NewYork, 1965).[2] Karlsson, P. W. Hypergeometric functions with integral parameter differences, J. Math.Phys. 12, 270–271, 1971.[3] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson’s theorem on thesum of a 3F2, Indian J. Math. 34, 23–32, 1992.[4] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Dixon’s theorem on the sumof a 3F2, Math. Comp. 62, 267–276, 1994.[5] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple’s theorem on thesum of a 3F2, J. Comput. Appl. Math. 72, 293–300, 1996.[6] Luke, Y. L. Mathematical Functions and Their Approximations (Academic Press, New York,1975).[7] Miller, A. R. Certain summation and transformation formulas for generalized hypergeomet-ric series, J. Comput. Appl. Math. 231, 964–972, 2009.[8] Miller, A. R. and Paris, R. B. Certain transformations and summations for the generalizedhypergeometric series with integral parameter differences, Integral Transforms and SpecialFunctions, 22, 67–77, 2011.[9] Miller, A. R. and Paris, R. B. Euler-type transformations for the generalized hypergeometricfunction r+2Fr+1(x), Zeitschrift angew. Math. Phys., 62, 31–45, 2011.[10] Miller, A. R. and Paris, R. B. On a result related to transformations and summations ofgeneralized hypergeometric series, Math. Communications, 17, 205–210, 2012.[11] Miller, A. R. and Paris, R. B. Transformation formulas for the generalized hypergeometricfunction with integral parameter differences, Rocky Mountain J. Math., 43, 291-327, 2013.[12] Miller, A. R. and Srivastava, H. M. Karlsson–Minton summation theorems for the gener-alized hypergeometric series of unit argument, Integral Transforms and Special Functions,21, 603–612, 2010.[13] Minton, B. M. Generalized hypergeometric function of unit argument, J. Math. Phys., 11,1375–1376, 1970.[14] Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. Integrals and Series: More SpecialFunctions, vol. 3 (Gordon and Breach, New York, 1990).[15] Rathie, A. K. and Pogány, T. K. New summation formula for 3F2(12) and a Kummer-typeII transformation of 2F2(x), Mathematical Communications, 13, 63–66, 2008.[16] Slater, L. J. Generalized Hypergeometric Functions (Cambridge University Press, Cam-bridge, 1966).

English

Some summation formulas for the hypergeometric series <sub>r+2</sub>F<sub>r+1</sub>(1/2)

Hacettepe Journal of Mathematics and StatisticsVolume 42 (3) (2013), 281 – 287SOME SUMMATION FORMULASFOR THE HYPERGEOMETRICSERIES r+2Fr+1(12)Y. S. Kim∗, A. K. Rathie†, U. Pandey‡ and R. B. Paris§ ¶Received 16 : 06 : 2011 : Accepted 22 : 06 : 2012AbstractThe aim of this paper is to obtain explicit expressions of the generalizedhypergeometric functionr+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]for j = 0,±1, . . . ,±5, where r pairs of numeratorial and denominatorialparameters differ by positive integers mr. The results are derived withthe help of an expansion in terms of a finite sum of 2F1(12) functions anda generalization of Gauss’ second summation theorem due to Lavoie etal. [J. Comput. Appl. Math. 72, 293–300 (1996)]. Some special andlimiting cases are also given.Keywords: Generalized hypergeometric series, Generalized Gauss summation theorem2000 AMS Classification: Primary 33C20, Secondary 33C15∗Department of Mathematics Education, Wonkwang University, Iksan, Korea.E-Mail: yspkim@wonkwang.ac.kr†Department of Mathematics, Central University of Kerala, Kasaragad 671328, Kerala, India.E-Mail: akrathie@rediffmail.com‡Department of Mathematics, Marudhar Engineering College, Raisar, Bikaner, RajasthanState, India. E-mail: pandey.up@rediffmail.com§University of Abertay Dundee, Dundee DD1 1HG, UK. E-Mail: r.paris@abertay.ac.uk¶Corresponding author282 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Paris1. IntroductionThe generalized hypergeometric function with p numeratorial and q denominatorialparameters is defined by the series [16, p. 41](1.1) pFq[a1, a2, . . . , apb1, b2, . . . , bq;x]=∞∑n=0(a1)n(a2)n . . . (ap)n(b1)n(b2)n . . . (bq)nxnn!,where (a)n = Γ(a+ n)/Γ(a) is the Pochhammer symbol (or ascending factorial). Whenq = p this series converges for |x| < ∞, but when q = p − 1 convergence occurs when|x| < 1 (unless the series terminates). In what follows we shall adopt the convention ofwriting the finite sequence of parameters (a1, . . . , ap) simply by (ap).It is well known that whenever hypergeometric functions reduce to gamma functions,the results are very important from the applications point of view. Thus, the classicaltheorems of Gauss, Kummer and Bailey for the series 2F1, and of Watson, Dixon, Whip-ple and Saalschu¨tz for the series 3F2, and others, play an important role in the theory ofhypergeometric and generalized hypergeometric series. In [3, 4, 5], Lavoie et al. consid-ered generalizations of some of the above-mentioned classical summations. In particular,they obtained a generalization of Gauss’ second summation theorem, given by2F1[a, b12a+ 12b+ 12; 12]= pi12Γ( 12a+ 12b+ 12)Γ( 12a+ 12)Γ( 12b+ 12)provided 12a+ 12b+ 126= 0,−1,−2, . . . , in the following form1.1. Theorem. [5] Provided 12(a+ b+ j + 1) 6= 0,−1,−2, . . ., we have the summation(1.2) 2F1[a, b12(a+ b+ j + 1); 12]= pi12Γ( 12a+ 12b+ 12j + 12)Γ( 12a− 12b− 12j + 12)Γ( 12a− 12b+ 12+ 12|j|)×{Aj(a, b)Γ( 12a+ 12)Γ( 12b+ δj+1)+Bj(a, b)Γ( 12a)Γ( 12b+ δj)}for integer j. As usual, [x] denotes the greatest integer less than or equal to x, its modulusis denoted by |x| and we have defined(1.3) δj ≡ 12 j − [ 12 j].The coefficients Aj(a, b) and Bj(a, b) are displayed in Table 1 for 0 ≤ j ≤ 5, where thecoefficients for j < 0 satisfy(1.4) A−j(a, b) = (−1)jAj(a, b), B−j(a, b) = (−1)j+1Bj(a, b).Table 1. The coefficients Aj and Bj for 0 ≤ j ≤ 5.j Aj(a, b) Bj(a, b)0 1 01 −1 12 12(a+ b− 1) −23 − 12(3a+ b− 2) 12(3b+ a− 2)4 14(a+ b− 1)(a+ b− 3) + ab −2(a+ b− 1)5 − 14(a+ b− 2)(5a+ b− 4) + a(b+ 1) 14(a+ b− 2)(5b+ a− 4) + b(a+ 1)Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 283Recently Miller [7], Miller and Paris [8, 9] and Miller and Srivastava [12] studiedthe generalized hypergeometric series (including integer parameter differences) and ob-tained numerous transformation and summation formulas for the series r+2Fr+1(z) andr+1Fr+1(z). In our present investigation, we shall require the following result establishedin [8] expressing r+2Fr+1(z), with r numeratorial and denominatorial parameters dif-fering by the positive integers (mr), in terms of a finite sum of Gauss hypergeometricfunctions.1.2. Theorem. [8] Let (mr) denote a set of positive integers with m = m1 + · · ·+mr.Then, when |z| < 1, we have(1.5) r+2Fr+1[a, b,c,(fr +mr)(fr); z]=m∑k=0(a)k(b)k(c)kzkCk(r) 2F1[a+ k, b+ kc+ k; z],where the coefficients Ck(r) are defined by [10](1.6) Ck(r) =(−1)kk!r+1Fr[−k, (fr +mr)(fr); 1].The expansion also holds when z = 1 provided Re (c− a− b) > m.Alternatively, the coefficients can be expressed in the form [8](1.7) Ck(r) ≡ 1Λm∑j=k{jk}σm−j , Λ = (f1)m1 . . . (fr)mr ,with C0(r) = 1, Cm(r) = 1/Λ, where {jk} denotes the Stirling number of the second kindand the σj (0 ≤ j ≤ m) are generated by the relation(1.8) (f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj .The result (1.5) and (1.6) can also be deduced as a particular case of the more generalexpansion given by Luke in [6, Eq. (5.10.2(4))] combined with the fact that Ck(r) = 0for k > m [2, 10].1.3. Remark. If we set z = 1 in (1.5), we immediately obtain the generalization of theKarlsson-Minton summation theorem [11, 12]r+2Fr+1[a, b,c,(fr +mr)(fr); 1]=Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)m∑k=0(−)k(a)k(b)kCk(r)(1 + a+ b− c)kprovided Re (c−a−b) > m. If c = b+1 and use is made of the alternative representationof the coefficients Ck(r) in (1.7), this reduces to the Karlsson-Minton summation formulagiven by [11, 13]r+2Fr+1[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)mrwhen Re (−a) > m− 1.Our aim in this paper is to obtain explicit expressions ofr+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]for complex parameters a, b and (fr) and for j = 0,±1, . . . ,±5, where r pairs of numera-torial and denominatorial parameters differ by positive integers (mr). For this purpose weshall make use of the expansion (1.5) combined with the generalization in (1.2) of Gauss’284 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Parissecond summation theorem. Several special and limiting cases of our main findings arealso presented.2. Summation formulasOur main result is given by the following theorem.2.1. Theorem. Let (mr) be a set of positive integers with m = m1 + · · · + mr and letj be an integer. Then, the generalized hypergeometric function of argument 12and withr pairs of numeratorial and denominatorial parameters differing by positive integers hasthe summation(2.1) r+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]= pi12Γ( 12a+ 12b+ 12j + 12)Γ( 12a− 12b− 12j + 12)Γ( 12a− 12b+ 12|j|+ 12)m∑k=0(a)k(b)kCk(r)(12)k×{Aj(a+ k, b+ k)Γ( 12a+ 12k + 12)Γ( 12b+ 12k + δj+1)+Bj(a+ k, b+ k)Γ( 12a+ 12k)Γ( 12b+ 12k + δj)},where the coefficients Aj and Bj for −5 ≤ j ≤ 5 can be obtained from the Aj(a, b) andBj(a, b) in Table 1 and use of (1.4) simply by putting a 7→ a + k and b 7→ b + k, and δjis defined in (1.3).Proof. If we set z = 12and c = 12(a+ b+ j + 1), where j is an integer, in (1.5) we obtainr+2Fr+1[a, b,12(a+ b+ j + 1),(fr +mr)(fr); 12]=m∑k=0(a)k(b)kCk(r)2k( 12a+ 12b+ 12j + 12)k2F1[a+ k, b+ k12a+ 12b+ 12j + 12+ k; 12].Application of (1.2) then leads to the result stated in (2.1). 2 To simplify the presentation of specific cases, we define the quantities D(1)± (k) andD(2)± (k) byD(1)± (k) ≡1a± b(Γ( 12a+ 12+ 12k)Γ( 12b+ 12k)± Γ( 12a+ 12k)Γ( 12b+ 12+ 12k)),D(2)± (k) ≡1(a± b)2 − 1((a+b+2k−1)Γ( 12a+ 12k)Γ( 12b+ 12k)± 4Γ( 12a+ 12+ 12k)Γ( 12b+ 12+ 12k)).Then, upon use of the duplication formula for the gamma function(2.2)√pi Γ(2z) = 22z−1Γ(z)Γ(z + 12),we have the following summations.2.2. Corollary. The summations in (2.1) for j = 0,±1,±2, respectively, take the form(2.3)r+2Fr+1[a, b,12(a+b+1),(fr +mr)(fr); 12]=2a+b−2√piΓ( 12a+ 12b+ 12)Γ(a)Γ(b)m∑k=02kCk(r) Γ(12a+ 12k)Γ( 12b+ 12k),Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 285(2.4)r+2Fr+1[a, b,12(a+b+1)± 12,(fr +mr)(fr); 12]=2a+b−1√piΓ( 12a+ 12b+1)Γ(a)Γ(b)m∑k=02kCk(r)D(1)∓ (k),(2.5)r+2Fr+1[a, b,12(a+b+1)± 1,(fr +mr)(fr); 12]=2a+b−1√piΓ( 12a+ 12b+ 32)Γ(a)Γ(b)m∑k=02kCk(r)D(2)∓ (k).When r = 0, the sequences (fr) and (fr +mr) are empty so that m = 0 and (2.3)–(2.5)reduce to (1.2).2.3. Remark. The results for j = −2, −1, 0 and 1 have been obtained by Miller andParis [8]. However, the results for j = −2 and j = −1 are given here in corrected forms.3. Specific examplesAs specific examples, we let r = 1, m1 = 1 and f1 = f , so that C0(1) = 1, C1(1) = 1/f ,and thus obtain(3.1) 3F2[a, b,12(a+ b+ 1)f + 1f; 12]= pi12 Γ( 12a+ 12b+ 12){1Γ( 12a+ 12)Γ( 12b+ 12)+2/fΓ( 12a)Γ( 12b)},(3.2) 3F2[a, b,12(a+ b+ 1)± 12,f + 1f; 12]= 2pi12Γ( 12a+ 12b+ 1)a∓ b{(a∓ f)/fΓ( 12a+ 12)Γ( 12b)∓ (b∓ f)/fΓ( 12a)Γ( 12b+ 12)},(3.3) 3F2[a, b,12(a+ b+ 1)± 1,f + 1f; 12]= 2pi12Γ( 12a+ 12b+ 32)(a∓ b)2 − 1{a+ b− 1∓ (2ab)/fΓ( 12a+ 12)Γ( 12b+ 12)∓ 4∓ 2(a+ b+ 1)/fΓ( 12a)Γ( 12b)}.3.1. Remark. The summation in (3.2) with the upper signs has been obtained previouslyin [10] and [15].The summations in (2.1) corresponding to j ≥ 1 are not valid when a−b = 0,±1, . . . ,±( 12j − 12) for odd j, and when a − b = ± 12, . . . ,±( 12j − 12) for even j. In such cases wemay use l’Hoˆpital’s rule and the duplication formula (2.2) to obtain the limiting casesgiven below.286 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Paris3.2. Corollary. When a = b, for example, we find the following summations correspond-ing to j = 1, 3 and 5 respectively:(3.4) r+2Fr+1[a, a,a+ 1,(fr +mr)(fr); 12](3.5) = 2a−1am∑k=0Ck(r)(a)k{ψ( 12a+ 12k + 12)− ψ( 12a+ 12k)},(3.6) r+2Fr+1[a, a,a+ 2,(fr +mr)(fr); 12]= 2a−1a(a+ 1)m∑k=0Ck(r)(a)k{(2a + 2k − 1){ψ( 12a + 12k)− ψ( 12a + 12k + 12)}+ 2}andr+2Fr+1[a, a,a+ 3,(fr +mr)(fr); 12]= 2a−1a(a+ 1)(a+ 2)m∑k=0Ck(r)(a)k{{(a+ k − 12)2 + 14}{ψ( 12a+ 12k + 12)− ψ( 12a+ 12k)} − (a+ k − 12)},where ψ is the digamma or psi function.3.3. Corollary. Similarly, if a− b = 1 when j = 2 and 4 we find the summations:(3.7) r+2Fr+1[a, a+ 1,a+ 2,(fr +mr)(fr); 12]= 2aa(a+ 1)m∑k=0Ck(r)(a+ 1)k{ψ( 12a+ 12k + 12)− ψ( 12a+ 12k)− 1a+ k}and(3.8) r+2Fr+1[a, a+ 1,a+ 3,(fr +mr)(fr); 12]= 2aa(a+ 1)(a+ 2)m∑k=0Ck(r)(a+ 1)k{(a+ k){ψ( 12a+ 12k)− ψ( 12a+ 12k + 12)}+ 1 + 12(a+ k)}.Finally, it is worthy of mention that when r = 0 we retrieve respectively from (3.4)–(3.8), or from (1.2), the following results for the series 2F1(12):2F1[a, aa+ 1; 12]= 2a−1a{ψ( 12a+ 12)− ψ( 12a)},2F1[a, aa+ 2; 12]= 2a−1a(a+ 1){(2a− 1){ψ( 12a)− ψ( 12a+ 12)}+ 2} ,2F1[a, aa+ 3; 12]= 2a−1a(a+ 1)(a+ 2){{(a− 12)2 + 14}{ψ( 12a+ 12)− ψ( 12a)} − (a− 12)},2F1[a, a+ 1a+ 2; 12]= 2aa(a+ 1){ψ( 12a+ 12)− ψ( 12a)− 1a}Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 287and2F1[a, a+ 1a+ 3; 12]= 2aa(a+ 1)(a+ 2){a{ψ( 12a)− ψ( 12a+ 12)}+ 1 + 12a}.3.4. Remark. The first of the above 2F1(12) summations is found in [1, p. 557] and in[14, p. 492] written in slightly different form, but the remaining results are believed tobe new.AcknowledgmentsOne of the authors (YSK) acknowledges the support of the Wonkwang UniversityResearch Fund (2012).References[1] Abramowitz, M. and Stegun, I. A. (Eds.) Handbook of Mathematical Functions (Dover, NewYork, 1965).[2] Karlsson, P. W. Hypergeometric functions with integral parameter differences, J. Math.Phys. 12, 270–271, 1971.[3] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson’s theorem on thesum of a 3F2, Indian J. Math. 34, 23–32, 1992.[4] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Dixon’s theorem on the sumof a 3F2, Math. Comp. 62, 267–276, 1994.[5] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple’s theorem on thesum of a 3F2, J. Comput. Appl. Math. 72, 293–300, 1996.[6] Luke, Y. L. Mathematical Functions and Their Approximations (Academic Press, New York,1975).[7] Miller, A. R. Certain summation and transformation formulas for generalized hypergeomet-ric series, J. Comput. Appl. Math. 231, 964–972, 2009.[8] Miller, A. R. and Paris, R. B. Certain transformations and summations for the generalizedhypergeometric series with integral parameter differences, Integral Transforms and SpecialFunctions, 22, 67–77, 2011.[9] Miller, A. R. and Paris, R. B. Euler-type transformations for the generalized hypergeometricfunction r+2Fr+1(x), Zeitschrift angew. Math. Phys., 62, 31–45, 2011.[10] Miller, A. R. and Paris, R. B. On a result related to transformations and summations ofgeneralized hypergeometric series, Math. Communications, 17, 205–210, 2012.[11] Miller, A. R. and Paris, R. B. Transformation formulas for the generalized hypergeometricfunction with integral parameter differences, Rocky Mountain J. Math., 43, 291-327, 2013.[12] Miller, A. R. and Srivastava, H. M. Karlsson–Minton summation theorems for the gener-alized hypergeometric series of unit argument, Integral Transforms and Special Functions,21, 603–612, 2010.[13] Minton, B. M. Generalized hypergeometric function of unit argument, J. Math. Phys., 11,1375–1376, 1970.[14] Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. Integrals and Series: More SpecialFunctions, vol. 3 (Gordon and Breach, New York, 1990).[15] Rathie, A. K. and Poga´ny, T. K. New summation formula for 3F2(12) and a Kummer-typeII transformation of 2F2(x), Mathematical Communications, 13, 63–66, 2008.[16] Slater, L. J. Generalized Hypergeometric Functions (Cambridge University Press, Cam-bridge, 1966).

https://rke.abertay.ac.uk/ws/files/8558523/Paris_SomeSummationFormulasForTheHypergeometricSeries_Published_2013.pdf

Some summation formulas for the hypergeometric series <sub>r+2</sub>F<sub>r+1</sub>(1/2)

Abstract

Similar works

Full text

Available Versions

Abertay Research Portal

Abertay Research Portal