# New Series Representations for Apéry’s and Other Classical Constants

## Abstract

We present a unified approach to obtain new series representations for various classical constants. Among others, we prove that

$$\log (2) = \frac{{17}}{{24}} + \sum\limits_{k = 2}^\infty {{{( - 1)}^k}} \frac{{{k^2} + k - 1/2}}{{(k - 1)k(k + 1)(k + 2)}}({H_k} - {H_{{{\left[ {k/2} \right]}}})^2}$$
$$G = - \frac{1}{2} + 2\sum\limits_{k = 1}^\infty {{{( - 1)}^k}\frac{{k(4{k^2} - 5)}}{{(4{k^2} - 1)(4{k^2} - 9)}}{{(2{H_{2k}} - {H_k})}^2}}$$

,

$$\zeta (3) = \frac{{149}}{{144}} + \frac{1}{8}\sum\limits_{k = 2}^\infty {\frac{{(2k + 1)({k^4} + 2{k^3} + 3{k^2} + 2k - 2)}}{{{{(k - 1)}^2}{k^2}{{(k + 1)}^2}(k + 2)}}{{(2H_k^{(2)} - H_{[k/2]}^{(2)})}^2}}$$

, where $${H_k} = \sum\nolimits_{j = 1}^k {1/j}$$ and $$H_k^{(2)} = {\sum\nolimits_{j = 1}^k {1/j} ^2}$$ denote the harmonic numbers and the generalized harmonic numbers of order 2, respectively, and G is the Catalan constant.

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## References

1. V. S. Adamchik, A certain series associated with Catalan’s constant, Z. Anal. Anwend., 21 (2003), 1–10.

2. J.-P. Allouche and J. Sondow, Infinite products with strongly B-multiplicative exponents, Ann. Univ. Sci. Budapest. Sec. Comput., 28 (2008), 35–53.

3. H. Alzer and J. Choi, The Riemann zeta function and classes of infinite series, Appl. Anal. Discr. Math., 11 (2017), 386–398.

4. H. Alzer, D. Karayannakis and H. M. Srivastava, Series representations for some mathematical constants, J. Math. Anal. Appl., 320 (2006), 145–162.

5. H. Alzer and S. Koumandos, Series representations for γ and other mathematical constants, Anal. Math., 34 (2008), 1–8.

6. H. Alzer and S. Koumandos, Series and product representations for some mathematical constants, Period. Math. Hungar., 58 (2009), 71–82.

7. H. Alzer and K. C. Richards, Series representations for special functions and mathematical constants, Ramanujan J., 40 (2016), 291–310.

8. H. Alzer and J. Sondow, A parameterized series representation for Apéry’s constant ζ(3), J. Comp. Anal. Appl., 20 (2016), 1380–1386.

9. J. M. Campbell and A. Sofo, An integral transform related to series involving alternating harmonic numbers, Int. Transf. Spec. Funct., 28 (2017), 547–559.

10. S. R. Finch, Mathematical Constants, Cambridge University Press (2003).

11. J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent, Ramanujan J., 16 (2008), 247–270.

12. W. Janous, Around Apéry’s constant, J. Inequal. Pure Appl. Math., 7 (2006), Article 35, 8 pp.

13. A. Sofo, Harmonic numbers at half-integer values, Int. Transf. Spec. Funct., 27 (2016), 430–442.

14. J. Sondow, Double-integrals for Euler’s constant and ln 4/π and an analog of Hadjicostas’s formula, Amer. Math. Monthly, 112 (2005), 61–65.

15. J. Sondow, A faster product for π and a new integral for ln π/2, Amer. Math. Monthly, 112 (2005), 729–734.

## Author information

Authors

### Corresponding author

Correspondence to H. Alzer.

Dedicated to the memory of Jean-Pierre Kahane

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Alzer, H., Sofo, A. New Series Representations for Apéry’s and Other Classical Constants. Anal Math 44, 287–297 (2018). https://doi.org/10.1007/s10476-018-0502-8

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• DOI: https://doi.org/10.1007/s10476-018-0502-8

### Key words and phrases

• series representation
• mathematical constant
• harmonic number

• 11A67