Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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DOUBLE SERIES TRANSFORMS DERIVED FROM FOURIER-LEGENDRE THEORY

  • Received : 2021.04.21
  • Accepted : 2021.09.23
  • Published : 2022.04.30
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Abstract

We apply Fourier-Legendre-based integration methods that had been given by Campbell in 2021, to evaluate new rational double hypergeometric sums involving ${\frac{{1}}{\pi}}$. Closed-form evaluations for dilogarithmic expressions are key to our proofs of these results. The single sums obtained from our double series are either inevaluable $_2F_1({\frac{4}{5}})$- or $_2F_1({\frac{1}{2}})$-series, or Ramanujan's 3F2(1)-series for the moments of the complete elliptic integral K. Furthermore, we make use of Ramanujan's finite sum identity for the aforementioned 3F2(1)-family to construct creative new proofs of Landau's asymptotic formula for the Landau constants.

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References

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