Abstract
By using various expansions of the digamma function and the method of residue computations, we study three variants of the linear Euler sums as well as related Hoffman’s double t-values and Kaneko-Tsumura’s double T-values and Kaneko-Tsumura’s double T-values and Kaneko–Tsumura’s double T-values, and establish several symmetric extensions of the Kaneko–Tsumura conjecture. Some special cases are discussed in detail to determine the coefficients of involved mathematical constants in the evaluations. In particular, it can be found that several general convolution identities on the classical Bernoulli numbers and Genocchi numbers are required in this study, and they are verified by the derivative polynomials of hyperbolic tangent.
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Acknowledgements
The authors would like to thank the anonymous referee for his (her) valuable comments and suggestions. The first author Weiping Wang is supported by the Zhejiang Provincial Natural Science Foundation of China (under Grant LY22A010018) and the National Natural Science Foundation of China (under Grant 11671360). The corresponding author Ce Xu is supported by the National Natural Science Foundation of China (under Grant 12101008), the Natural Science Foundation of Anhui Province (under Grant 2108085QA01) and the University Natural Science Research Project of Anhui Province (under Grant KJ2020A0057).
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Wang, W., Xu, C. On variants of the Euler sums and symmetric extensions of the Kaneko–Tsumura conjecture. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 84 (2023). https://doi.org/10.1007/s13398-023-01398-7
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DOI: https://doi.org/10.1007/s13398-023-01398-7
Keywords
- Multiple zeta values
- Multiple t-values
- Multiple T-values
- Harmonic numbers
- Euler sums
- Convolution identities
- Bernoulli numbers
- Genocchi numbers