Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers

Transcription

1 Quadratic Trasformatios of Hypergeometric Fuctio ad Series with Harmoic Numbers Marti Nicholso I this brief ote, we show how to apply Kummer s ad other quadratic trasformatio formulas for Gauss ad geeralized hypergeometric fuctios i order to obtai trasformatio ad summatio formulas for series with harmoic umbers that cotai oe or two cotiuous parameters. There is a extesive literature o applicatio of Newto-Adrews method for fidig idetities with harmoic umbers H = k ad geeralized Harmoic umbers H r = For a geeral descriptio of this method ad partial overview of the literature oe ca cosult the papers [ 4]. The aim of this paper is to study trasformatio formulas that are obtaied from quadratic trasformatios of hypergeometric fuctio by itegratio usig Euler s itegral represetatio for Harmoic umbers z H = dz. z k= k= k r. We will use the stadard otatio for hypergeometric fuctio a,... a r rf s ; z = b,..., b s = where a = aa +... a + is the Pochhammer symbol. a... a r!b... b s z Theorem. Let a ad b be arbitrary complex umbers such that a+b / is ot a egative iteger. The 4 a b H! a + b + / = a b H,! a + b + / a b H + H! a + b + / = a b! a + b + / H + H. Proof. Startig from the quadratic trasformatio formula.. i [] a, b a, b F a + b + ; z = F a + b + ; 4z z, Re z <, replace z i equatio with z ad multiply both sides by zc to get a b! a + b + / zc z = a b! a + b + / z c z z,

2 where ow z [, ]. This series coverges uiformly. Next we itegrate this series termwise with respect to z from to. The itegral o the LHS is easy to calculate ad equals Bc, +. The itegral o the RHS is z c z z dz = = = z c z dz z c z dz z c z dz = B c+, +. After multiplicatio of both sides by c we deduce the series trasformatio a b Γc + Γ +! a + b + / = Γc + + a b Γ c + Γ +! a + b + / Γ c + +. Next differetiate both sides of this equatio with respect to c at c = usig formulas d Γc + Γ + dc Γc + + = H, c= d Γc + Γ + dc Γc + + = H + H c= from which the results stated i the theorem follow immediately. Corollary. Let ψx = Γ x Γx a a! deote the digamma fuctio. The H = Proof. Put b = a i the first formula of Theorem, π Γ aγ + aψ a + ψ + a ψ ψ. a a! H = I view of Gauss summatio formula [], formula.8.46 we have as required. a a! H. a a! H = d a, dc F a c + c= ; = d Γc + Γc + dc Γc a + Γc + a c= π = Γ aγ + aψ a + ψ + a ψ ψ

3 Examples. 4 H 3 = Γ l π π 3!! 3 H 54 = Γ3 3 7/3 π H 8 = Γ 3 9 l 3 π 8 Γ π 6 l π H 3 = Γ l π π 3!! 3 H 3 54 = Γ3 3 7/3 π H + H 3 = Γ 4 4π 3/ 3 + l 3 l 3 π C π 4 + l π l Here C is Catala s Costat. Series that cotai both cetral biomial coefficiets ad Harmoic umbers are studied i [5 8], ad the series where additioally the cetral biomial coefficiets are squared have bee studied i [9 ]. Formulas 3-5 are direct cosequeces of the Corollary. To derive 6 ad 7 observe that differetiatig Kummer s summatio formula [],.8.5 a, b F a + b + ; with respect to b yields the summatio formula [3] a b! a + b + from which by specializig a = b = 4 k= Γ a + b + Γ a + = Γ a + k a + b + + k oe obtais the sum 4H 3H 3 = Γ 4 6 l 4. π π = Γ Γ a + b + Γ a +, Together with 3 this allows to calculate the sum 6. Formula 7 is derived i aalogous maer ad the proof of 8 is similar to the proof of Corollary. Theorem. Let Rea + b <. The a b! a + b + / H + = a + b si πa si πb ψ a b cos πa + b + ψ 3 a b ψ a ψ b

4 4 Proof. Cosider formula..7 i [] Γ Γa + b + a, b Γa + Γb + F ; z We take the differece at z = ad at z Γ Γa + b + Γa + Γb + a b! / z = a, b = F a + b + ; + z a, b + F a + b + ; z. a b! a + b + / the divide by z ad itegrate. The itegral o the LHS is z z dz = k + = H H, k= z + z, while the itegral o the RHS Thus z + z dz z = = a b H = Γ Γ a + b +! a + b + / Γ a + z t dt = t H. dz + z a b! / H H. To compute the sum o the RHS oe ca differetiate Gauss summatio formula [], formula.8.46 a, b F c ; ΓcΓc a b = Γc aγc b 9 with respect to c at c = to obtai a b H H = Γ Γ a b ψ! / Γ aγ b + ψ a b ψ a ψ b. After simplifyig the product of Gamma fuctios usig Euler s reflectio formula, ad after chage of parameters a a, b b oe recovers the formula stated i the theorem. Note. Summatio formulas of the same type as i Theorem were foud i [4] by applyig Newto- Adrews method to aalogs of Watso s sum a, b, c 3F a + b +, c; = Γ Γa + b + Γc + Γ a b + c Γa + Γb + Γ a + cγ b + c. Two of such aalogs ca be compactly writte as a, b, c + ε 3F a + b +, c; = Γ ΓcΓ a + b + Γc a b Γ a + Γc aγc b ΓcΓ a + b + + ε Γ Γ a b + c ΓaΓbΓ a + c + Γ b + c +, ε = ±.

5 Formula with ε = is due to Wei [4]. Below we give a sketch of a alterative proof of Theorem usig Newto-Adrews method followig the approach i [4]. Differetiatig with respect to c at c = yields the sum = a b! a + b + / + + H + i terms of gamma ad digamma fuctios. The first term i the brackets is summed by with c =. The sum with the secod term ca be expressed as = a b a + b =! a + b + / + a b lim a, b, c 3F c c a + b, c + ;. Now the hypergeometric fuctio is calculated usig with ε =, therefore it is possible to express this limit i terms of gamma fuctios ad its derivatives. Hece the sum i Theorem is expressible i terms of gamma ad digamma fuctios, as required. Theorem 3. Let Re a + b >, the / a + b + a + b H = 4 a b, /, a + b H + 3 F + a + b + a, + b ; l 4. Proof. I the quadratic trasformatio formula 4.5. from [] a, b, c 3F a b +, a c + ; z = + z a3f a b c +, a, a + 4z ; a b +, a c + + z we put a = ad the take the differece at z = ad at z to obtai b c b c z = / b c b c [ + z 4z + z After dividig by z ad itegratig termwise we get the followig itegral o the RHS [ ] z dz + z + z z = + t t dt t + t = t dt t + t As a result b c b c H = 4 = 4 H l 4. Oe ca easily brig this to the symmetric form stated i the theorem. Theorem 4. Let Re b >, the 4 / b! b H = / b c b c H l 4. / b H + Γ b + Γb! b + / ΓbΓ b l. ]. 5

6 6 Proof. I the formula..5 from [] a, b + z a F b ; 4z a, a + + z = F b b + ; z we put a = ad the cosider its differece at z = ad at z / b! b [ ] 4z / b + z + z = z.! b + / After dividig by z ad itegratig with respect to z oe obtais H o the RHS, while the itegral o the LHS was calculated i the proof of theorem 3. Thus / b! b H / b! b l 4 = 4 / b! b + / H. Evaluatig the secod series usig Gauss sum 9 completes the proof. Whe b is a positive iteger Theorems 3 ad 4 allow oe to express the value of a ifiite sum with Harmoic umbers as a fiite sum. Ackowledgemets. The author of this paper wish to thak Dr. Wechag Chu for valuable correspodece ad ecouragemet. [] A. Erdelyi, W. Magus, F. Oberhettiger ad F.G. Tricomi, Higher Trascedetal Fuctios, Vol. I. McGraw- Hill Book Compay, Ic., New York-Toroto-Lodo 953. [] W.C. Chu, L. De Doo, Hypergeometric series ad harmoic umber idetities, Adv. Appl. Math [3] J. Choi, H.M. Srivastava, Some summatio formulas ivolvig harmoic umbers ad geeralized harmoic umbers, Mathematical ad Computer Modellig, 54, -34. [4] W. Wag, C. Jia, Harmoic umber idetities via the Newto-Adrews method, Ramauja J. 35, [5] M.Yu. Kalmykov, B.F.L. Ward, S.A. Yost, Multiple Iverse biomial sums of arbitrary weight ad depth ad the all-order ε-expasio of geeralized hypergeometric fuctios with oe half-iteger value of parameter, J. High Eergy Phys., [6] K.N. Boyadzhiev, Series with cetral biomial coefficiets, Catala umbers, ad harmoic umbers, J. Iteger Seq. 5, Article..7,. [7] R. Tauraso, Some Cogrueces for Cetral Biomial Sums Ivolvig Fiboacci ad Lucas Numbers, J. Iteger Seq. 9, Article [8] W. Chu, D. Zheg, Ifiite series with harmoic umbers ad cetral biomial coefficiets, Iteratioal Joural of Number Theory, 5, [9] H. Che, Iterestig series associated with cetral biomial coefficiets, Catala umbers ad harmoic umbers, J. Iteger Seq. 9, Article [] H. Liu, W. Zhou, S. Dig, Geeralized harmoic umber summatio formulae via hypergeometric series ad digamma fuctios, Joural of Differece Equatios ad Applicatios 3, [] J.M. Campbell, A. Sofo, A itegral trasform related to series ivolvig alteratig harmoic umbers, Itegral Trasforms ad Special Fuctios, 8, [] J.M. Campbell, Ramauja-like series for π ivolvig harmoic umbers 7. upublished [3] J. Choi, H.M. Srivastava, Certai Classes of Ifiite Series, Mh. Math. 7, [4] C. Wei, Watso-type 3F -series ad summatio formulae ivolvig geeralized harmoic umbers, arxiv: