A new proof of quadratic series of Au-Yeung and explicit evaluation of its alternating sum

Narendra Bhandari () and Yogesh Joshi ()
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Narendra Bhandari: Rajdhani College, University of Delhi
Yogesh Joshi: Rajdhani College, University of Delhi

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1251-1260

Abstract: Abstract We revisit the quadratic series of Au-Yeung $$\sum _{n=1}^{\infty }\left( \frac{H_n}{n}\right) ^2$$ ∑ n = 1 ∞ H n n 2 , which is quite well-known in the mathematical literature, and we consider its alternating sum $$\sum _{n=1}^{\infty }(-1)^{n+1}\left( \frac{H_n}{n}\right) ^2.$$ ∑ n = 1 ∞ ( - 1 ) n + 1 H n n 2 . The central notion of this paper is to address a new approach to the famous quadratic series of Au-Yeung via the construction of a few classes of logarithmic integrals with the tails of the dilogarithm functions, which on computation leads to the elementary harmonic sums and polylogarithm sums. We also give an explicit proof of its alternating sum.

Keywords: Harmonic number; Euler sum; Logarithmic integral; Dilogarithm function; Abel’s summation formula; Alternating Euler sum; 40C10; 33B30; 40G10 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00431-0

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