Abstract
Various methods are used to investigate sums involving a reciprocal central binomial coefficient and a power term. In the first part, new functions are introduced for calculation of sums with a negative exponent in the power term. A recurrence equation for the functions provides an integral representation of the sums using polylogarithm functions. Thus polylogarithms and, in particular, zeta values can be expressed via these functions, too. In the second part, a straightforward recurrence formula is derived for sums having a positive exponent in the power term. Finally, two interesting cases of double sums are presented.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and main results
The subject of this article are infinite sums, \(A^{+}_m\) and \(A^{-}_m\), involving a reciprocal central binomial coefficient \(1/{2k\atopwithdelims ()k}\) and a power term \(k^{-m}\):
In the context of these sums—which we also call the Apéry sums—Roger Apéry proved the irrationality of \(\pi ^2\) and \(\zeta (3)\) [1, 9, 13]. Furthermore, the tight relationship between these sums and values of the zeta function are the topic of many publications [2, 6,7,8, 10,11,12,13,14].
Whereas there exist closed-form expressions for \(A^{+}_1,A^{+}_2,A^{-}_1,A^{-}_2\), and as well \(A^{+}_4\), see for instance [10,11,12,13], the sums \(A^{+}_m\) and \(A^{-}_m\) for \(m>4\) can only be expressed by means of polylogarithmic ladders or multiple Clausen values [2, 7]. In particular,
where \(\phi =(\sqrt{5}+1)/2\), see e.g. [2, 11,12,13].
In this work two approaches are presented to calculate the sums with a negative exponent in the power term, \(A^{+}_m\) and \(A^{-}_m\) for \(m>0\). First, using a reformulation of the reciprocal central binomial coefficient we define the functions \(A^{+}_m(x)\) and \(A^{-}_m(x)\):
These functions \(A^{+}_m(x)\) and \(A^{-}_m(x)\) together with their derivatives \(A'^{+}_m(x)\) and \(A'^{-}_m(x)\) relate the Apéry sums \(A^{+}_m\) and \(A^{-}_m\) with the polylogarithm functions \(\mathrm{Li}_{\,n}(x)\), defined as
Theorem 1.1
\( A^{+}_m = A^{+}_m(1) \), \( A^{-}_m = A^{-}_m(1) \).
Theorem 1.2
For the proofs of Theorems 1.1 and 1.2 see (11) and Lemma 2.1 in Sect. 2.
Thus, integrating (5) the Apéry sums for \(m>0\) can be calculated as
Furthermore, as shown in Sect. 2, the polylogarithm \(\mathrm{Li}_{\,m}(z)\), and in particular its value \(\mathrm{Li}_{\,m}(1)=\zeta (m)\), can be expressed vice versa in terms of \(A_m(x)\).
In Sect. 3 we derive an integral representation of the reciprocal central binomial coefficient, which gives us a direct integral formula of the sums, see (6). In Sect. 4 we calculate the integrals for \(m=1,2,3\).
Due to the exponential convergence of the reciprocal binomial coefficients even the sums with a positive exponent in the power term, \(A^{+}_{-m}\) and \(A^{-}_{-m}\) for \(m\geqslant 0\), converge. These sums up to \(m=3\) are studied in various articles [11, 12, 15]. In this work a straightforward recurrence formula is presented to calculate them.
Theorem 1.3
For the proof see (26) in Sect. 5. By reiterative application of (7), the sums \(A^{+}_{-m}\) and \(A^{-}_{-m}\) are calculated as linear combinations of 1 and \(A^{+}_1\), respectively 1 and \(A^{-}_1\), where \(A^{+}_1=\sqrt{3}\pi /9\) and \(A^{-}_1=2\ln \phi /{\sqrt{5}}\), see (2).
Theorem 1.4
The numbers \(3^mA^{+}_{-m}\) and \(5^mA^{-}_{-m}\) are sums of an integer and odd—and therefore non-vanishing—multiple of \(2A^{+}_1\), respectively \(2A^{-}_1\).
For the proof see Lemma 5.1 in Sect. 5. Since \(\pi \) and (due to the Gelfond–Schneider theorem [3,4,5]) \(\ln \phi \) are transcendental, \(A^{+}_1\) and \(A^{-}_1\) are transcendental, too. Then, due to Theorem 1.4, \(A^{+}_{-m}\) and \(A^{-}_{-m}\) for all \(m\geqslant 0\) are transcendental numbers. For instance, this is different from \(A^{-}_3\) which is proven only to be irrational [1].
Although there is up to now—with the exception of \(A^{+}_4\)—no closed-form expression for \(A^{+}_m\) and \(A^{-}_m\) for all \(m\geqslant 3\), the sums of \(A^{+}_m-{1}/{2}\) and \(A^{-}_m-{1}/{2}\) over m are calculated as follows.
Theorem 1.5
For the proof see (30) in Sect. 6. Finally, in Sect. 6 the identities, \(0= 6A^{+}_0A^{+}_2-4A^{+}_1A^{+}_2-3A^{+}_1A^{+}_1\) and \(0=10A^{-}_0A^{-}_2-4A^{-}_1A^{-}_2-5A^{-}_1A^{-}_1\), are presented as vanishing double sums over the products \(1/\bigl [{2k\atopwithdelims ()k}{2j\atopwithdelims ()j}\bigr ]\).
2 Recurrence equation
We show below that non-alternating sums \(A^{+}_m\) and alternating sums \(A^{-}_m\) subordinate, in general, to the same equations, except for a sign. First, we simplify the notation: , , and . Reiterative application of the recurrence formula for reciprocal binomial coefficients,
to the reciprocal central binomial coefficient \(1/{2k\atopwithdelims ()k}\) gives
Thus the sums \(A_m\), (1), for \(m\geqslant 1\), can be reformulated via coefficients \(\alpha _{mjk}\) as
Now, for a more detailed investigation of the sums \(A_m\) the functions \(A_m(x)\) are introduced, see (3):
In the end of this section the coefficients \(a^{+}_{1j}\) and \(a^{-}_{1j}\) are calculated. Convergence of the sums \(A^{+}_m(x)\) and \(A^{-}_m(x)\) is guaranteed for \(|x|<1\), respectively \(|x|<1/\phi \), only. However, by the process of analytic continuation the order of summation in (10) can be switched and a comparison with (9) shows
The derivatives \(A'_m(x)\) and \(A''_m(x)\) are given by
For \(m=1\) the derivative \(A'_1(x)\)—as the analytic continuation of (12)—is calculated as
In fact, the coefficients \(a_{mj}\) from (10) satisfy the identity
Thus, for the functions \(A_m(x)\) from (10) the following differential equation is obtained:
Therefore, the derivatives \(A'_m(x)\) can be calculated recurrently:
Since \(A'_{m-1}(0)=a_{m1}=0\), by integration \(A'_m(x)\) can be written as
Lemma 2.1
The function \((x-1)A'_m(x)\) and the polylogarithm \({{\mp }} \mathrm{Li}_{\,m-1}({\mp }(x^2-x))\) are identical:
Proof
(i) For \(m=1\) both sides are identical and (ii) for \(m>1\) they are described by the same recurrence equation.
(i) In the case \(m=1\), it follows from (13) and the definition \(\mathrm{Li}_0(z)=z/(1-z)\):
(ii) Applying (15) to the recurrence equation of polylogarithm, \(\mathrm{Li}_{\,m-1}'(z) =\mathrm{Li}_{\,m-2}(z)/z\) with \(z={\mp }(x^2-x)\), results in (14), again:
\(\square \)
Due to the symmetry of (15),
and the transformation, \(x\mapsto {\frac{1}{2}}-x\), the Apéry sums \(A_m=A_m(1)\) are calculated as
Integrating (16), the polylogarithm \({\mp }\mathrm{Li}_m({\mp }(x-x^2))\) can be described by the function \(A_m(x)\):
in which \(\mathrm{Li}_m(0)=A_m(0)=0\) and the symmetry of (17) are used.
The values of the zeta function are given by \(\zeta (m)=\mathrm{Li}_{\,m}(1)\). Using the definition of the coefficients \(a^{+}_{mj}\) from (9), and writing with \(\varphi -\varphi ^2=1\) and \(1-\varphi =\varphi ^{-1}\), one obtains
Thus, the functions \(A^{+}_m(x)\), or respectively the coefficients \(a^{+}_{mj}\), enable a formulation for both the sums \(A^{+}_m\) and the values \(\zeta (m)\).
Next, we want to focus on the coefficients \(a_{mj}\) from (9) in the case \(m=1\). Here, the coefficients \(a^{+}_{1j}\) of the non-alternating sum \(A^{+}_1(x)\) fulfil the equation
With \(a^{+}_{11}=0\) and \(a^{+}_{12}=1\) one gets recurrently,
Hence, comparing the sum \(A^{+}_1=A^{+}_1(1)\) from (10) with (2) gives
The coefficients \(a^{-}_{1j}\) of the alternating sum \(A^{-}_1(x)\) can be written as
With \(a^{-}_{11}=0\) and \(a^{-}_{12}=1\) the coefficients \(a^{-}_{1,j+1}\) are given by the Fibonacci numbers \(F_j\):
Since Fibonacci numbers grow exponentially with \(\lim _{j\rightarrow \infty }F_j/F_{j-1}=\phi \), at \(x=1\) the sum \(A^{-}_1(1)=\sum _{j=1}^\infty (-1)^ja^{-}_{1j}/j\) does not converge.
Having \(|a^{+}_{1j}|\leqslant 1\) for all j, the convergence radius of \(A^{+}_1(x)\), (10), is \(R^+_1=1\). The convergence radius of \(A^{-}_1(x)\) is given by \(R^-_1=\lim _{j\rightarrow \infty }|a^{-}_{1,j-1}/a^{-}_{1j}|=1/\phi \). Furthermore, with \(|a^{+}_{mj}|\leqslant |a^{+}_{m-1,j}|\) and \(|a^{-}_{mj}|\leqslant |a^{-}_{m-1,j}|\) it follows \(R^+_m\geqslant 1\) and \(R^-_m\geqslant 1/\phi \).
3 Integral representation
The integrals \(A_m\) can be obtained directly by the integral representation of the reciprocal central binomial coefficients \(1/{2k\atopwithdelims ()k}\), see e.g. [15]. Looking at the integrals \(\int _0^{1/2}\bigl ({\frac{1}{4}}-x^2\bigr ){}^{k-1} \mathrm{d}x\), partial integration results in
Thus, starting with \(\int _0^{1/2} \bigl ({\frac{1}{4}}-x^2\bigr )^0 \mathrm{d}x = \frac{1}{2}\), we arrive to the integral by induction:
So, the reciprocal central binomial coefficients \(1/{2k\atopwithdelims ()k}\) are
With (8) and the definition of the polylogarithm function, \(\mathrm{Li}_{\,m}(z)=\sum _{k=1}^\infty k^{-m}z^k\), the sums \(A_m\) can be calculated as
in agreement with (18). A similar result was derived by Taylor [15].
Furthermore, calculating the integral formula for the coefficients \(1/{2k\atopwithdelims ()k}\) in (20) one obtains the identity
4 Integrals for \(A_1\), \(A_2\), and \(A_3\)
The calculation of the sums \(A_m\) for \(m>0\), and in particular \(A_1\) and \(A_2\), see (2), has been considered in many publications, see e.g. [11,12,13]. In this section we employ the integrals from (18) to calculate \(A_1\) and \(A_2\) and to give an integral formula of \(A_3\). For \(m=1\) the integral \(A_1\) from (18) is given by \(\mathrm{Li}_0(z)=z/(1-z)\):
In particular, \(A^{+}_1\) and \(A^{-}_1\) are calculated as
For \(m=2\) the integral \(A_2\) from (18) can be calculated via :
Having the factorizations,
the integral , the dilogarithm \(\mathrm{Li}_2(0)=0\), and the Landen identities,
\(A^{+}_2\) and \(A^{-}_2\) are obtained as
The integral \(A_3\) from (18) is calculated by integration of the dilogarithm \(\mathrm{Li}_2\). Integrating by parts one finds
Thus the sums \(A^{+}_3\) and \(A^{-}_3={\frac{2}{5}}\zeta (3)\), see (2), are given by the following integrals:
leading to another integral form for \(\zeta (3)\).
5 Apéry sums for positive powers
The sums for positive powers, \(A_{-m}\) for \(m\geqslant 0\), have also been studied in [11, 12, 15]. Calculation of \(A_{0},A_{-1},A_{-2}\), and \(A_{-3}\) was done by Lehmer [12]. In this section, a general recurrence formula for \(A_{-m}\) is derived. The sums \(A_{-m}\) converge, since \(1/{2k\atopwithdelims ()k}<{\frac{k}{4^k}}\) and
With (1) the expression \(4 A_m - 2 A_{m+1}\) can be calculated by
Thus, we obtain in the case of positive powers, with \(m\geqslant 0\), that
This directly leads to the recurrence equation for the sums \(A_{-m}\):
Here, the sums \(A_{-m}\) can be calculated recurrently starting with \(A_1\). For \(0 \leqslant m \leqslant 4\) the sums \(A^{+}_{-m}\) and \(A^{-}_{-m}\) are given by:
Using the substitution \({\widetilde{A}}_{-m} = (4 {\mp } 1)^{m+1} A_{-m}\), (26) gives
Here, \(w_{mj}\) are the weights of \({\widetilde{A}}_{-j}\) contributing to \({\widetilde{A}}_{-m}\).
Lemma 5.1
The numbers \({\widetilde{A}}_{-m}\) are sums of an integer and an odd multiple of \(2 A_1\).
Proof
First we prove \({\widetilde{A}}_0\) is a sum of an integer and an odd multiple of \(2 A_1\). All weights \(w_{mj}\) are integer numbers and the sum of weights,
is an odd number. Then, by induction, each number \({\widetilde{A}}_{-m}\) is a sum of an integer and an odd multiple of \(2 A_1\). \(\square \)
6 Further relations on Apéry sums
In the same way as the zeta function \(\zeta (m)=\sum _{k=1}^\infty k^{-m}\) and eta function \(\eta (m)=\sum _{k=1}^\infty (-1)^{k-1}k^{-m}\) converge to 1 for \(m\rightarrow \infty \), the sums \(A_m=\sum _{k=1}^\infty (\pm 1)^{k-1}k^{-m}/{2k\atopwithdelims ()k}\) converge to \(\frac{1}{2}\), since all terms for \(k\geqslant 2\) vanish when \(m\rightarrow \infty \), and the term for \(k=1\) establishes the limit. The sum of Apéry sums, \(S=\sum _{m=1}^\infty \bigl (A_m-{\frac{1}{2}}\bigr )\), can be written as
To calculate S, first we construct the sum of \(4A_m-2A_{m+1}-1\), (25), over m:
Regrouping (29) leads to
Comparing one gets
Thus, the sum \(S^{+}\) of the non-alternating sums \(A^{+}_m\) and the sum \(S^{-}\) of the alternating sums \(A^{-}_m\) are given by
Again, these equations resemble similar equations for the zeta and eta functions, namely \(\sum _{m=2}^\infty (\zeta (m)-1)=1\) and \(\sum _{m=2}^\infty (\eta (m)-1)=1-2\eta (1)=1-2\ln 2\).
Finally, the sums \(A^{+}_1\) and \(A^{+}_2\) from (21) and (23), and also \(A^{-}_1\) and \(A^{-}_2\) of (22) and (24), are related by
In addition, the relations between \(A^{+}_0\) and \(A^{+}_1\), and also between \(A^{-}_0\) and \(A^{-}_1\), are given by (27) and (28):
Thus, combining (31) and (32), the following double sums are shown to be 0:
Here, the question remains open whether these equations could be proved in rational terms, i.e., without an explicit calculation of the irrational sums \(A_1\) and \(A_2\).
References
Apéry, R.: Irrationalité de \(\zeta (2)\) et \(\zeta (3)\). Astérisque 61, 11–13 (1979)
Bailey, D.H., Borwein, J.M., Bradley, D.M.: Experimental determination of Apéry-like identities for \(\zeta (2n+2)\). Experiment. Math. 15(3), 281–289 (2006)
Baker, A.: Linear forms in the logarithms of algebraic numbers. I. Mathematika 13, 204–216 (1966)
Baker, A.: Linear forms in the logarithms of algebraic numbers. II. Mathematika 14, 102–107 (1967)
Baker, A.: Linear forms in the logarithms of algebraic numbers. III. Mathematika 14, 220–228 (1967)
Beukers, F.: A note on the irrationality of \(\zeta (2)\) and \(\zeta (3)\). Bull. London Math. Soc. 11(3), 268–272 (1979)
Borwein, J.M., Broadhurst, D.J., Kamnitzer, J.: Central binomial sums, multiple Clausen values and zeta values. Experiment. Math. 10(1), 25–34 (2001)
Borwein, J.M., Bradley, D.M.: Empirically determined Apéry-like formulae for \(\zeta \)(4n+3). Experiment. Math. 6(3), 181–194 (1997)
Cohen, H.: Démonstration de l’irrationalité de \(\zeta (3)\) (d’apres R. Apery). Séminaire de Théorie des Nombres de Grenoble, 6, # 6 (1978)
Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974)
Finch, S.R.: Mathematical Constants II. §1.22 Central binomial coefficients. Encyclopedia of Mathematics and its Applications, vol. 169, pp. 182–188. Cambridge University Press, Cambridge (2019)
Lehmer, D.H.: Interesting series involving the central binomial coefficient. Amer. Math. Monthly 92(7), 449–457 (1985)
van der Poorten, A.: A proof that Euler missed. Math. Intell. 1(4), 195–203 (1978)
Sprugnoli, R.: Sums of reciprocals of the central binomial coefficients. Integers 6, # A27 (2006)
Taylor, R.: Integrals and interesting series involving the central binomial coefficients (2013). https://web.williams.edu/Mathematics/sjmiller/public_html/hudson/TaylorR_IntSeries.pdf
Acknowledgements
I am grateful to Jörn Steuding for his kind support and helpful suggestions. I would also like to thank the referees for a number of valuable improvements and for correcting some errors.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Uhl, M. Recurrence equation and integral representation of Apéry sums. European Journal of Mathematics 7, 793–806 (2021). https://doi.org/10.1007/s40879-020-00415-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-020-00415-y