Certain transformation and summation formulas for generalized hypergeometric series with integral parameter differences are derived

Miller, A. R.

Paris, Richard B.

Integral Transforms and Special Functions

English

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Certain transformations and summations for generalizedhypergeometric series with integral parameter differencesA. R. Miller†Formerly Professor of Mathematics at George Washington University,1616 18th Street NW, No. 210, Washington, DC 20009-2525, USAandR. B. ParisDivision of Complex Systems,University of Abertay Dundee, Dundee DD1 1HG, UKr.paris@abertay.ac.ukAbstractCertain transformation and summation formulas for generalized hypergeometric serieswith integral parameter differences are derived.Mathematics Subject Classification: 33C15, 33C20Keywords: Generalized hypergeometric series, Quadratic transformation, Summation the-orems1. IntroductionThe generalized hypergeometric function pFq(x) is defined for complex parameters and argumentby the seriespFq(a1, a2, . . . , apb1, b2, . . . , bq∣∣∣∣x)=∞∑k=0(a1)k(a2)k . . . (ap)k(b1)k(b2)k . . . (bq)kxkk!. (1.1)When q = p this series converges for |x| < ∞, but when q = p − 1 convergence occurs when|x| < 1. However, when only one of the numeratorial parameters aj is a negative integer or zero,then the series always converges since it is simply a polynomial in x of degree −aj . In (1.1)the Pochhammer symbol, or ascending factorial, (a)k is defined by (a)0 = 1 and for k ≥ 1 by(a)k = a(a+ 1) . . . (a+ k − 1). However, for all integers k we write simply(a)k =Γ(a+ k)Γ(a).In what follows we shall adopt the convention of writing the finite (except where noted otherwise)sequence of parameters (a1, . . . , ap) simply by (ap) and the product of p Pochhammer symbols by((ap))k ≡ (a1)k . . . (ap)k,with an empty product p = 0 reducing to unity.In [1–4] we derived in various ways transformation formulas for the generalized hypergeometricfunctions r+1Fr+1(x) and r+2Fr+1(x), where here and below at least r pairs of numeratorial and1denominatorial parameters differ by arbitrary positive integers. In particular, in [4] we statedwithout proof that the generalized hypergeometric function r+1Fr+1(x) in which r + 1 pairs ofnumeratorial and denominatorial parameters differ by arbitrary positive integers may be writtenas a product of ex and a certain polynomial in x. In Section 3 we shall provide a proof of thisresult and discuss its implications.In [1, 4, 5] we showed that essentially the same methods used to obtain the transformation for-mulas alluded to above may be employed to deduce the Karlsson-Minton and other more generalsummation formulas for the r+2Fr+1(1) generalized hypergeometric series with unit argument. Inthe present investigation we shall derive in Section 4 in a similar manner analogous summationformulas for generalized hypergeometric series of the type r+2Fr+1(12) with half unit argument.Finally, in Section 5 we shall consider a certain quadratic transformation for the 3F2(x) hyper-geometric function.2. Preliminary resultsWe record two lemmas and a theorem that we shall utilize in the sequel. Lemmas 1 and 2are proved respectively in [1] and [4] and Theorem 1 is proved in [4]. The notation {nk} will beemployed to denote the Stirling numbers of the second kind. These nonnegative integers representthe number of ways to partition n objects into k nonempty sets and arise for nonnegative integersn in the generating relation [6, p. 262]xn =n∑k=0{nk}(−1)k(−x)k,{n0}= δ0n,where δ0n is the Kronecker symbol.Lemma 1. For nonnegative integers j defineSj ≡∞∑n=0njλnn!, S0 ≡∞∑n=0λnn!,where the infinite sequence (λn) is such that Sj converges for all j. ThenSj =j∑k=0{jk} ∞∑n=0λn+kn!.Lemma 2. For nonnegative integer s let (as) denote a parameter sequence containing s elements,where when s = 0 the sequence is empty. Let (as +k) denote the sequence when k is added to eachelement of (as). Let F(x) denote the generalized hypergeometric function with r numeratorialand denominatorial parameters differing by the positive integers (mr), namelyF(x) ≡ r+sFr+1((as),c,(fr +mr)(fr)∣∣∣∣x),where by (1.1) 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 , A0 = (f1)m1 . . . (fr)mr , Am = 1 (2.1)2and the σj (0 ≤ j ≤ m) are generated by the relation(f1 + x)m1 · · · (fr + x)mr =m∑j=0σm−jxj . (2.2)Theorem 1. Let (mr) be a nonempty sequence of positive integers and define m ≡ m1+· · ·+mr.Then if b 6= fj (1 ≤ j ≤ r), (λ)m 6= 0, where λ ≡ c− b−m, we have the transformation formular+1Fr+1(b,c,(fr +mr)(fr)∣∣∣∣x)= ex m+1Fm+1(λ,c,(ξm + 1)(ξm)∣∣∣∣− x), (2.3)where |x| < ∞. The (ξm) are the nonvanishing zeros of the associated parametric polynomialQm(t) of degree m given byQm(t) =m∑j=0σm−jj∑k=0{jk}(b)k(t)k(λ− t)m−k,where the σj (0 ≤ j ≤ m) are determined by the generating relation (2.2).In the following Section 3 we shall consider the generalized hypergeometric function w(x)defined for |x| <∞ byw(x) ≡ r+1Fr+1((fr+1 +mr+1)(fr+1)∣∣∣∣x),where (mr+1) is a sequence of positive integers. It is evident that w(x) is an entire function.3. Properties of w(x)If in Theorem 1 we set b = fr+1 +mr+1, c = fr+1 and define m ≡ m1 + · · ·+mr, M ≡ m+mr+1,then λ = −M , (−M)m 6= 0 and we find from (2.3) thatw(x) = ex m+1Fm+1(−M,fr+1,(ξm + 1)(ξm)∣∣∣∣− x).The (ξm) are the nonvanishing zeros of the associated parametric polynomial of degree m givenbyQm(t) =m∑j=0σm−jj∑k=0{jk}(fr+1 +mr+1)k(t)k(−M − t)m−k,where the σj (0 ≤ j ≤ m) are generated by (2.2).Thus it is evident that w(x) is proportional to a polynomial in x of degree at most M whichwe define asPM (x) ≡ m+1Fm+1(−M,fr+1,(ξm + 1)(ξm)∣∣∣∣− x).Moreover, since ex can never vanish it follows that the entire function w(x) has at most M zerosin the complex plane. However, we shall obtain an explicit representation for PM (x) which showsthat its degree is exactly M .Theorem 2. Let (mr+1) be a sequence of positive integers such that M ≡ m1 + · · · + mr+1and let (fr+1) be a sequence of complex numbers such that (f1)m1 . . . (fr+1)mr+1 is nonvanishing.Thenw(x) = exPM (x),3where PM (x) is a polynomial of degree M given byPM (x) =1B0M∑k=0Bkxk, Bk ≡M∑j=k{jk}ρM-j.HereB0 = (f1)m1 . . . (fr+1)mr+1 , BM = 1,and the ρj (0 ≤ j ≤M) are generated by the relation(f1 + x)m1 . . . (fr+1 + x)mr+1 =M∑j=0ρM-jxj .Proof: Note that((fr+1 +mr+1))n((fr+1))n=(f1 + n)m1(f1)m1. . .(fr+1 + n)mr+1(fr)mr+1,where the numeratorial expression on the right-hand side of the latter may be written as(f1 + n)m1 . . . (fr+1 + n)mr+1 =M∑j=0ρM-jnj . (3.1)Thus by (1.1)w(x) ≡ r+1Fr+1((fr+1 +mr+1)(fr+1)∣∣∣∣x)=∞∑n=0((fr+1 +mr+1))n((fr+1))nxnn!=1(f1)m1 . . . (fr+1)mr+1M∑j=0ρM-j∞∑n=0njxnn!,where the order of the summations has been interchanged.Now employing Lemma 1 we have∞∑n=0njxnn!=j∑k=0{jk} ∞∑n=0xn+kn!= exj∑k=0{jk}xkso thatw(x) =ex(f1)m1 . . . (fr+1)mr+1M∑j=0ρM-jj∑k=0{jk}xk,whereM∑j=0ρM-jj∑k=0{jk}xk =M∑k=0M∑j=k{jk}ρM-jxk.DefiningBk ≡M∑j=k{jk}ρM-j(0 ≤ k ≤M),we note that when k = 0 the only contribution to the j-summation comes from j = 0, so that byusing (3.1) we findB0 = ρM = (f1)m1 . . . (fr+1)mr+1 .4In addition when k = M , the only contribution to the latter summation comes from j = M , sothat by again using (3.1) we see thatBM = ρ0 = 1.Thusw(x) =exB0M∑k=0Bkxk (3.2)which evidently completes the proof. 2Thus we also have the following.Corollary 1. The entire function w(x) has exactly M zeros in the complex plane.We remark that Ki and Kim [7] only show the existence of at most M zeros for w(x), whereasfrom (3.2) we can in principle obtain all of the M zeros. For example,2F2(f + 1,f,g + 1g∣∣∣∣x)= ex(1fgx2 +f + g + 1fgx+ 1),and here the zeros of P2(x) arex1,2 = − 12 (f + g + 1 ± [(f − g)2 + 2f + 2g + 1]1/2).4. Summation formulas for r+2Fr+1(12)In [4] we employed Lemma 2 to readily obtain a generalization of the Karlsson-Minton summationformula which is given in the following.Theorem 3. Suppose (mr) is a sequence of positive integers such that m = m1 + · · ·+mr. Thenprovided 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=0AkA0(−1)k(a)k(b)k(1 + a+ b− c)k,where the Ak (0 ≤ k ≤ m) are given by (2.1).However we may also utilize Lemma 2 to obtain summation formulas for r+2Fr+1(12). To thisend we note the following (see, for example, [8, Section 7.3.7 (3)–(6)]) summations for 2F1(12)given by2F1(a, b12a+ 12b− 12∣∣∣∣12)=√π{Γ(12a+ 12b+ 12)Γ(12a+ 12)Γ(12b+ 12)+2Γ(12a+ 12b− 12)Γ(a)Γ(b)}, (4.1)2F1(a, b12a+ 12b∣∣∣∣12)=√π{Γ(12a+ 12b)Γ(12b)Γ(12a+ 12)+Γ(12a+ 12b)Γ(12a)Γ(12b+ 12)}, (4.2)2F1(a, b12a+ 12b+ 12∣∣∣∣12)=√πΓ(12a+ 12b+ 12)Γ(12a+ 12)Γ(12b+ 12), (4.3)2F1(a, b12a+ 12b+1∣∣∣∣12)=2√πa− b{Γ(12a+ 12b+1)Γ(12a)Γ(12b+ 12)− Γ(12a+ 12b+1)Γ(12b)Γ(12a+ 12)}. (4.4)5Setting x = 12, s = 2, (as) = (a, b), c =12(a + b + n), where n = −1, 0, 1, 2, in the first twoequations of Lemma 2, we see thatr+2Fr+1(a, b,12a+ 12b+ 12n,(fr +mr)(fr)∣∣∣∣12)=m∑k=0(12)kAkA0(a)k(b)k(12a+ 12b+ 12n)k2F1(a+ k, b+ k12a+ 12b+ 12n+ k∣∣∣∣12), (4.5)where when r = 0, (fr) and (fr +mr) are empty and we define m ≡ 0. Thus the Gauss series2F1(12) in (4.5) with respectively n = −1, 0, 1, 2 correspond to the left-hand sides of (4.1)–(4.4)with a 7→ a+ k and b 7→ b + k. Combining these with (4.5) we therefore deduce the following.Theorem 4. Let (mr) be an arbitrary sequence of positive integers such that m ≡ m1 + · · ·+mrand (fr) a sequence of complex numbers such that (f1)m1 . . . (fr)mr 6= 0. Thenr+2Fr+1(a, b,12a+ 12b− 12,(fr +mr)(fr)∣∣∣∣12)=2√πA0Γ(12a+ 12b− 12)Γ(a)Γ(b)m∑k=0(12)kAk{1 +Γ(a+k)Γ(b+k)(a+b−1+2k)4Γ(12a+ 12+ 12k)Γ(12b+ 12+ 12k)}, (4.6)r+2Fr+1(a, b,12a+ 12b,(fr +mr)(fr)∣∣∣∣12)=2a+b−2A0√πΓ(12a+ 12b)Γ(a)Γ(b)m∑k=02kAk{Γ(12a+ 12k)Γ(12b+ 12+ 12k) − Γ(12b+ 12k)Γ(12a+ 12+ 12k)}, (4.7)r+2Fr+1(a, b,12a+ 12b+ 12,(fr +mr)(fr)∣∣∣∣12)=2a+b−2A0√πΓ(12a+ 12b+ 12)Γ(a)Γ(b)m∑k=02kAk Γ(12a+ 12k)Γ(12b+ 12k), (4.8)r+2Fr+1(a, b,12a+ 12b+1,(fr +mr)(fr)∣∣∣∣12)=2a+b−2A0√πa+ ba− bΓ(12a+ 12b)Γ(a)Γ(b)m∑k=02kAk{Γ(12b+ 12k)Γ(12a+ 12k+ 12) − Γ(12a+ 12k)Γ(12b+ 12k+ 12)},(4.9)where the Ak (0 ≤ k ≤ m) are given by (2.1).We have used the duplication formula√π Γ(2z) = 22z−1Γ(z)Γ(z + 12) (4.10)to obtain (4.7)–(4.9). Thus, when r = 0, the sequences (fr) and (fr + mr) are empty so thatm = 0 and (4.6)–(4.9) reduce respectively to (4.1)–(4.4). Since (4.9) is not valid when a = b wemay use l’Hôpital’s rule and (4.10) to obtain the limiting case of (4.9), namelyr+2Fr+1(a, a,a+ 1,(fr +mr)(fr)∣∣∣∣12)=2a−1aA0m∑k=0Ak(a)k{ψ(12a+ 12k+ 12) − ψ(12a+ 12k)},6where ψ is the digamma or psi function. When r = 0 we retrieve2F1(a, aa+ 1∣∣∣∣12)= 2a−1a{ψ(12a+ 12) − ψ(12a)}which is found in [8, Section 7.3.7 (16)] in an equivalent form.In addition, if we let r = 1, m1 = 1 and f1 = f , so that A0 = f , A1 = 1, then we obtain from(4.9) and (4.10) after some straightforward algebra the result (when a 6= b)3F2(a, b,12a+ 12b+1,f + 1f∣∣∣∣12)=√π(a+ ba− b)Γ(12a+ 12b){(a− f)/fΓ(12b)Γ(12a+ 12)− (b − f)/fΓ(12a)Γ(12b+ 12)},which has been obtained by Rathie and Pogány in [9].5. A quadratic transformation for 3F2(x)In [4] by utilizing Lemma 2 we derived two quadratic transformations forr+2Fr+1(a, a+ 12,c,(fr +mr)(fr)∣∣∣∣X),where X ≡ x2/(1 ± x)2 and X ≡ 4x/(1 + x)2. As discussed below certain parametric conditionsguaranteeing the existence of a quadratic transformation for the Gauss function 2F1(x) restrictthe existence of more general quadratic transformations for r+2Fr+1(x) when the decompositionLemma 2 is employed. We illustrate this by deducing a quadratic transformation that necessarilyis restricted to r = 1 thus giving the quadratic transformation for 3F2(x) in the following theorem.Theorem 5. Suppose a, b 6= 12and f 6= a+ b+ 12. Then3F2(a, b,a+ b+ 12,f + 1f∣∣∣∣4x(1 − x))= (1 − 2x)−1 4F3(2a− 1, 2b− 1,a+ b+ 12,ξ1 + 1,ξ1,ξ2 + 1ξ2∣∣∣∣x), (5.1)where ξ1, ξ2 are given byξ1,2 = α+12± [(α+ 12)2 − 2αf ]1/2, α = (2a− 1)(2b− 1)2(a+ b− f) − 1 . (5.2)The transformation (5.1) holds in a neighborhood of x = 0.Proof: From Lemma 2 we have the expansion of r+2Fr+1(z), with r pairs of parameters differingby the positive integers (mr), as a finite sum of 2F1(z) functions in the formr+2Fr+1(a, b,c,(fr +mr)(fr)∣∣∣∣z)=1A0m∑k=0zkAk(a)k(b)k(c)k2F1(a+ k, b+ kc+ k∣∣∣∣z), (5.3)where the coefficients Ak are defined by (2.1). If we set r = 1, m1 = 1 and f1 = f in the aboveexpansion we obtain the particular case3F2(a, b,c,f + 1f∣∣∣∣z)= 2F1(a, bc∣∣∣∣z)+abzcf2F1(a+ 1, b+ 1c+ 1∣∣∣∣z). (5.4)7Now let z = X ≡ 4x(1 − x) and defineF (x) ≡ 3F2(a, b,a+ b+ 12,f + 1f∣∣∣∣X).Then, from (5.4) with c = a+ b+ 12, we findF (x) = 2F1(a, ba+ b+ 12∣∣∣∣X)+abcfX 2F1(a+ 1, b+ 1a+ b+ 32∣∣∣∣X). (5.5)We now make use of two quadratic transformation formulas for the Gauss hypergeometric function[10, Section 15.3, (22) and (24)], namely2F1(a, ba+ b− j + 12∣∣∣∣X)= (1 − 2x)−j 2F1(2a− j, 2b− ja+ b− j + 12∣∣∣∣x), (5.6)where j = 0, 1. Application of these transformations with j = 0 and j = 1 respectively to thefirst and second hypergeometric functions in (5.5) then leads toF (x) = (1 − 2x)−1{(1 − 2x) 2F1(2a, 2ba+ b+ 12∣∣∣∣x)+4abcfx(1 − x) 2F1(2a+ 1, 2b+ 1a+ b+ 32∣∣∣∣x)}. (5.7)From (1.1) it can be seen that for arbitrary values of c2F1(2a, 2bc∣∣∣∣x)=∞∑n=0(2a)n(2b)n(c)nxnn!=∞∑n=0(2a− 1)n(2b− 1)n(c)nxnn!(2a− 1 + n)(2b− 1 + n)(2a− 1)(2b− 1) ,x 2F1(2a, 2bc∣∣∣∣x)=∞∑n=1(2a)n−1(2b)n−1(c)n−1xn(n− 1)!=∞∑n=0(2a− 1)n(2b− 1)n(c)nxnn!n(c− 1 + n)(2a− 1)(2b− 1) ,x 2F1(2a+ 1, 2b+ 1c+ 1∣∣∣∣x)=∞∑n=1(2a+ 1)n−1(2b+ 1)n−1(c+ 1)n−1xn(n− 1)!=c4ab∞∑n=0(2a− 1)n(2b− 1)n(c)nxnn!n(2a− 1 + n)(2b− 1 + n)(2a− 1)(2b− 1) ,andx22F1(2a+ 1, 2b+ 1c+ 1∣∣∣∣x)=∞∑n=2(2a+ 1)n−2(2b+ 1)n−2(c+ 1)n−2xn(n− 2)!=c4ab∞∑n=0(2a− 1)n(2b− 1)n(c)nxnn!n(n− 1)(c− 1 + n)(2a− 1)(2b− 1) ,8where obvious adjustments to the summation index n have been made. Then, upon settingc = a+ b+ 12in the above sums, we obtain from (5.7)F (x) =(1 − 2x)−1(2a− 1)(2b− 1)∞∑n=0(2a− 1)n(2b− 1)n(a+ b+ 12)nxnn!P2(n),where, with γ ≡ f−1(a+ b− 12) − 1,P2(n) = (2a− 1 + n)(2b− 1 + n)(1 +nf)− n(2 +n− 1f)(a+ b− 12+ n)= γn2 +(γ +1f(2a− 1)(2b− 1))n+ (2a− 1)(2b− 1). (5.8)The quadratic P2(n) can be factored to yieldP2(n) = γ(n+ ξ1)(n+ ξ2),where, provided a, b 6= 12and f 6= a + b − 12, ξ1 and ξ2 are given by (5.2). Since γξ1ξ2 =(2a− 1)(2b− 1), we therefore see thatF (x) = (1 − 2x)−1∞∑n=0(2a− 1)n(2b− 1)n(a+ b+ 12)nxnn!(n+ ξ1)(n+ ξ2)ξ1 ξ2= (1 − 2x)−1∞∑n=0(2a− 1)n(2b− 1)n(a+ b+ 12)n(ξ1 + 1)n(ξ2 + 1)n(ξ1)n (ξ2)nxnn!,where we have made use of the fact that (λ + 1)n/(λ)n = (n + λ)/λ. This yields (5.1) and socompletes the proof. 2We remark that when f = a+ b− 12, the coefficient γ = 0 and the polynomial P2(n) becomeslinear with ξ1 = f . The formula (5.1) then reduces to the quadratic transformation (5.6) withj = 1.If we try to apply the same reasoning to the more general function with r pairs of parametersdiffering by the positive integers (mr)r+2Fr+1(a, b,a+ b+ 12,(fr +mr)(fr)∣∣∣∣X), (5.9)we obtain from (5.3) a series of m+ 1 terms involving the Gauss functionsFk ≡ 2F1(a+ k, b+ ka+ b+ k + 12∣∣∣∣X)(0 ≤ k ≤ m).A quadratic transformation for 2F1(α, β; γ |x) exists if and only if any of the quantities±(1 − γ), ±(α− β), ±(α+ β − γ)are such that either one of them equals 12or two of them are equal [10, p. 560]. In this case, thethird condition above for the functions Fk, with 0 ≤ k ≤ m, has the form α+β− γ = k− 12 ; thatis, a quadratic transformation exists only when k = 0 and k = 1. Consequently, we are compelledto take r = 1, m = 1 in (5.9).9References[1] A. R. Miller, Certain summation and transformation formulas for generalized hypergeometric series, J. Com-put. Appl. Math. 231 (2009) 964–972.[2] A. R. Miller and R. B. Paris, A generalized Kummer-type transformation for the pFp(x) hypergeometricfunction, Canadian Math. Bulletin (to appear 2011).[3] A. R. Miller and R. B. Paris, Euler-type transformations for the generalized hypergeometric functionr+2Fr+1(x) (submitted for publication 2010).[4] A. R. Miller and R. B. Paris, Transformation formulas for the generalized hypergeometric function withintegral parameter differences (submitted for publication 2010).[5] A. R. Miller and H. M. Srivastava, Karlsson-Minton summation theorems for the generalized hypergeometricseries of unit argument, Integral Transforms and Special Functions (in press 2010).[6] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second ed., Addison-Wesley, UpperSaddle River, 1994.[7] H. Ki and Y.-O. Kim, On the zeros of some generalized hypergeometric functions, J. Math. Anal. Appl. 243(2000) 249–260.[8] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, vol. 3, Gordon and Breach, NewYork, 1990.[9] A. K. Rathie and T. K. Pogány, New summation formula for 3F2(12) and a Kummer-type II transformationof 2F2(x), Mathematical Communications 13 (2008) 63–66.[10] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.10

Certain transformations and summations for generalized hypergeometric series with integral parameter differences

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Certain transformations and summations for generalized hypergeometric series with integral parameter differences

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