Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers

Transcription

1 Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers Martin Nicholson In this brief note, we show how to apply Kummer s and other quadratic transformation formulas for Gauss and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. There is an extensive literature on application of Newton-Andrews method for finding identities with harmonic numbers n H n = k and generalized Harmonic numbers H r n = For a general description of this method and partial overview of the literature one can consult the papers [3 5]. The aim of this paper is to study transformation formulas that are obtained from quadratic transformations of hypergeometric function by integration using Euler s integral representation for Harmonic numbers z n H n = dz. z k= n k= k r. We will use the standard notation for hypergeometric function a,... a r rf s ; z = b,..., b s n= where a n = aa +... a + n is the Pochhammer symbol. a n... a r n n!b n... b s n z n Theorem. Let a and b be arbitrary complex numbers such that a+b / is not a negative integer. Then a n b n H n n! a + b + / n n = a n b n H n, n! a + b + / n a n b n Hn + H n n! a + b + / n n = a n b n n! a + b + / n H n + H n. Proof. Starting from the quadratic transformation formula.. in [] a, b a, b F a + b + ; z = F a + b + ; z z, Re z <, replace z in equation with z and multiply both sides by zc to get a n b n n! a + b + / n n zc z n = a n b n n! a + b + / n z c z z n,

2 where now z [, ]. This series converges uniformly. Next we integrate this series termwise with respect to z from to. The integral on the LHS is easy to calculate and equals Bc, n +. The integral on the RHS is z c z z n dz = = = z c z n dz z c z n dz z c z n dz = B c+, n +. After multiplication of both sides by c we deduce the series transformation a n b n Γc + Γn + n! a + b + / n n = Γc + n + a n b n Γ c + Γn + n! a + b + / n Γ c + n +. Next differentiate both sides of this equation with respect to c at c = using formulas d Γc + Γn + dc Γc + n + = H n, c= d Γc + Γn + dc Γc + n + = Hn + H n, c= from which the results stated in the theorem follow immediately. Corollary. Let ψx = Γ x Γx denote the digamma function. Then Proof. In the first formula of Theorem put b = The sum on the left equals n + a n a + 3 n H n n = a + ψa + ψ. a n n + H n a + 3/ n n = a n H n. 3 a + b + / n a n H n = d a, c a + 3/ n dc F a + 3 ; = d Γa + 3 Γ 3 c dc Γ 3 Γa + 3 c After combining 3 and the proof is complete. Corollary. With the same notations as in Corollary we have a n a n H n π n! n = c= c= = a + ψa + ψ. Γ a a+ Γ ψ a + ψ a+ ψ ψ.

3 Proof. This formula may be deduced from Theorem as a corollary, but the easiest way to prove it is by differentiating the summation formula.8.5 in [] with respect to c at c =. Examples. a, a F c + ; = Γ c + Γ c+ Γ c a + Γ c+a+ n H n n 3 n = Γ ln π π 3n! n! 3 H n 5 n = Γ3 3 7/3 π 3 9 ln 3 π n H n n 3 n = Γ 8 3 ln π π 3n! n! 3 H 3n 5 n = Γ3 3 7/3 π 3 + ln 3 ln 3 π Series that contain both central binomial coefficients and Harmonic numbers are studied in [6 9], and the series where additionally the central binomial coefficients are squared or cubed have been studied in [ 7]. Formulas 5,6 are direct consequences of the Corollary. To derive 7 and 8 observe that differentiating Kummer s summation formula [],.8.5 a, b F a + b + ; with respect to b yields the summation formula [8] a n b n n n n! a + b + from which by specializing a = b = n k= Γ a + b + Γ a + Γ b + = Γ b + k a + b + + k one obtains the sum n H n 3H n n 3 n = Γ 6 ln. π π = Γ Γ a + b + Γ a + Γ b +, Together with 5 this allows to calculate the sum 7. Formula 8 is derived in analogous manner. Note that the series n H n n 6 n kn = K k + π Kk log k 6 k, n H n n 6 n kn = K k + π Kk log k k have closed form in terms of logarithms and complete elliptic integrals of the first kind see [] for a summation equivalent to a certain linear combination of the two formulas above

4 Theorem. The following generalization of Corollary holds a n a n n! H n x n π a, a = sin πa F ; x + ψ a a+ + ψ ψ ψ π sin πa log x x a, a F ; x. 9 Proof. The hypergeometric function F a, b; c; x satisfies the differential equation [], section.. x xu + c a + b + xu abu =. Let vx = d dc F a, a; c; x c=. Then vx satisfies the differential equation x xv + xv a av = fx, where fx = d dx F a, a; ; x. Note that vx is the series on the left side of 9. The homogeneous equation with fx = has two linearly independent solutions F a, a; ; x and F a, a; ; x. One can check by direct substitution for example using computer algebra systems that partial solution of the non-homogeneous equation is Thus the general solution of equation is v x = F a, a; ; x log x x. vx = v x + A F a, a; ; x + B F a, a; ; x. The coefficients A and B are determined from two conditions v = v = ψ a a+ a, a + ψ ψ ψ F where the second condition is the Corollary. The first condition may be replaced by the weaker condition of the cancellation of the logarithmic singularity at x +. Due to the asymptotics sin πa F a, a; ; x = log π x + O, the first condition implies that B = π sin πa. A is now easily determined from the second condition. It is known that when a = r, r =, 3,, 6 the hypergeometric function F a, a; ; x is an elliptic integral [], ch. 33. As a result both F r, r ; ; x and F r, r ; ; x have closed form in terms of gamma functions when x are certain algebraic numbers, for example F 3, 3 ; ; 33 3 = 3 F 3, 3 ; ; ;, = 3 3/8 + 3 / Γ π 3/. Thus Theorem allows one to generate closed form summation formulas with Harmonic numbers. Also note that for r =, 3,, 6 the expression n /r /r n n! takes the values n, 3n n 6 n n 7 n n n, n n 6 n n n, 6n 3n 3 n 3n n. It is also worth mentioning the summation formula 3n! H 3n n! 3 7 n xn = π 3 3 F 3, 3 ; x F 3, 3 3 x ; x log 6. x Generating functions similar to 9 but with a sum of digamma functions instead of H n are considered in [9] as a special case of more general formulas from [], ch..3..

5 5 Theorem 3. Let Rea + b <. Then a n b n n! a + b + / n H n n + = Proof. Consider formula..7 in [] Γ Γa + b + a, b Γa + Γb + F a + b sin πa sin πb ψ a b cos πa + b + ψ 3 a b ψ a ψ b. ; z We take the difference at z = and at z Γ Γa + b + Γa + Γb + a n b n n! / n z n = a, b = F a + b + ; + z a, b + F a + b + ; z. a n b n n! a + b + / n then divide by z and integrate. The integral on the LHS is while the integral on the RHS Thus z n + z z n n z dz = k + = H n H n, k= n dz z = = a n b n H n = Γ Γ a + b + n! a + b + / n Γ a + Γ b + z n + z z n t n dt = t H n. dz + z a n b n n! / n H n H n. n, To compute the sum on the RHS one can differentiate Gauss summation formula [], formula.8.6 a, b F c ; ΓcΓc a b = Γc aγc b with respect to c at c = to obtain a n b n H n H n = Γ Γ a b ψ n! / n Γ aγ b + ψ a b ψ a ψ b. After simplifying the product of Gamma functions using Euler s reflection formula, and after change of parameters a a, b b one recovers the formula stated in the theorem. Summation formulas of the same type as in Theorem 3 were found in [] by applying Newton-Andrews method to analogs of Watson s sum a, b, c 3F a + b +, c; = Γ Γa + b + Γc + Γ a b + c Γa + Γb + Γ a + cγ 3 b + c.

6 Two of such analogs can be compactly written as [] a, b, c + ε 3F a + b +, c; = Γ ΓcΓ a + b + Γc a b Γ a + Γ b + Γc aγc b ΓcΓ a + b + + ε Γ Γ a b + c ΓaΓbΓ a + c + Γ b + c +, ε = ±. Below we give a sketch of an alternative proof of Theorem 3 using Newton-Andrews method. Differentiating 3 with respect to c at c = yields the sum a n b n n! a + b + / n n + n + H n n + n= n= in terms of gamma and digamma functions. The first term in the brackets is summed by 3 with c =. The sum with the second term can be expressed as a n b n a + b = n! a + b + / n n + a b lim a, b, c 3F c c a + b., c + ; Now the hypergeometric function is calculated using with ε =, therefore it is possible to express this limit in terms of gamma functions and its derivatives. Hence the sum in Theorem 3 is expressible in terms of gamma and digamma functions, as required. Theorem. Let Re a + b >, then / n a + b n + a n + b n H n = a n b n, /, a + b n H n + 3 F + a n + b n + a, + b ; ln. Proof. In the quadratic transformation formula.5. from [] a, b, c 3F a b +, a c + ; z = + z a3f a b c +, a, a + z ; a b +, a c + + z we put a = and then take the difference at z = and at z to obtain b n c n b n c n n z n = / n b c n b n c n [ + z z + z After dividing by z and integrating termwise we get the following integral on the RHS [ ] z n dz + z + z z = + t t n dt t + t = t n dt t + t As a result b n c n b n c n n H n = = H n ln. One can easily bring this to the symmetric form stated in the theorem. / n b c n b n c n H n ln. ] n. 6

7 7 Theorem 5. Let Re b >, then / n b n n! b n H n = Proof. In the formula..5 from [] / n b n H n + Γ b + Γb n! b + / n ΓbΓ b ln. a, b + z a F b ; z a, a + + z = F b b + ; z we put a = and then consider its difference at z = and at z / n b n n! b n [ ] z n + z + z = / n b n n! b + / n z n. After dividing by z and integrating with respect to z one obtains H n on the RHS, while the integral on the LHS was calculated in the proof of theorem. Thus / n b n n! b n H n / n b n n! b n ln = / n b n n! b + / n H n. Evaluating the second series using Gauss sum completes the proof. When b is a positive integer Theorems and 5 allow one to express the value of an infinite sum with Harmonic numbers as a finite sum. Acknowledgements. The author of this paper wish to thank Dr. Wenchang Chu for valuable correspondence and encouragement. References [] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. I. McGraw- Hill Book Company, Inc., New York-Toronto-London 953. [] B.C. Berndt, Ramanujan s Notebooks, Part V, Springer, New York 998. [3] W.C. Chu, L. De Donno, Hypergeometric series and harmonic number identities, Adv. Appl. Math [] J. Choi, H.M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Mathematical and Computer Modelling, 5, -3. [5] W. Wang, C. Jia, Harmonic number identities via the Newton-Andrews method, Ramanujan J. 35, [6] M.Yu. Kalmykov, B.F.L. Ward, S.A. Yost, Multiple Inverse binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter, J. High Energy Phys., 8 7. [7] K.N. Boyadzhiev, Series with central binomial coefficients, Catalan numbers, and harmonic numbers, J. Integer Seq. 5, Article..7,. [8] R. Tauraso, Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers, J. Integer Seq. 9, Article [9] W. Chu, D. Zheng, Infinite series with harmonic numbers and central binomial coefficients, International Journal of Number Theory, 5, [] M.W. Coffey, On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions, J. Math. Phys. 9, 35 8.

8 [] S. Boettner, V.H. Moll, The integrals in Gradshteyn and Ryzhik. Part 6: Complete elliptic integrals, arxiv:5.9. [] H. Chen, Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers, J. Integer Seq. 9, Article [3] H. Liu, W. Zhou, S. Ding, Generalized harmonic number summation formulae via hypergeometric series and digamma functions, Journal of Difference Equations and Applications 3, [] J.M. Campbell, A. Sofo, An integral transform related to series involving alternating harmonic numbers, Integral Transforms and Special Functions, 8, [5] J.M. Campbell, Ramanujan-like series for π involving harmonic numbers 7. [6] J. Guillera, More hypergeometric identities related to Ramanujan-type series, Ramanujan J. 3, 5-3. [7] J.G. Wan, Random Walks, Elliptic Integrals and Related Constants, PhD Thesis, 3. [8] J. Choi, H.M. Srivastava, Certain Classes of Infinite Series, Mh. Math. 7, [9] D. Borwein, J.M. Borwein, M.L. Glasser, J.G. Wan, Moments of Ramanujan s generalized elliptic integrals and extensions of Catalan s constant, J. Math. Anal. Appl., 38, [] C. Wei, Watson-type 3F -series and summation formulae involving generalized harmonic numbers, arxiv: [] S. Lewanowicz, Generalized Watson s summation formula for 3 F, J. Comput. Appl. Math. 86,