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PreprintCC BY 4.0arXiv

Discrete Hyperbolic Secant Distributions

  • Leonard Pleschberger
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Abstract

We introduce a family of discrete hyperbolic secant distributions on ℤ, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant G=ϖ/π, where ϖ=Γ2(1/4)/(2√(2π)) is the lemniscate constant. Using the elliptic lambda-star function λ*, we construct scaled versions of these distributions parametrized by √r and 1/√r for r ∈ ℕ. For the first such distribution, we compute the moments up to the eighth degree in closed form via Poisson summation, exploiting that the hyperbolic secant is a fixed point of the −2πi-Fourier transform. As a byproduct, we obtain closed-form values for series of odd powers of the hyperbolic secant up to the ninth degree, e.\,g.\ ∑k ∈ ℤ sech3(πk) = (G3+G)/√2, and reinterpret several classical Ramanujan series probabilistically. Finally, we apply our results to evaluate a contour integral, on the critical line, of the product of Dirichlet's beta, gamma, and Riemann zeta functions.

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Cite

BibTeXRIS
BibTeX
@misc{pleschberger2026discrete,
  title = {{Discrete Hyperbolic Secant Distributions}},
  author = {Leonard Pleschberger},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11293},
  eprint = {2609.11293},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11293},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Discrete Hyperbolic Secant Distributions
AU  - Leonard Pleschberger
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11293
UR  - https://arxiv.org/abs/2609.11293
AB  - We introduce a family of discrete hyperbolic secant distributions on ℤ, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant G=ϖ/π, where ϖ=Γ^2(1/4)/(2√(2π)) is the lemniscate constant. Using the elliptic lambda-star function λ^*, we construct scaled versions of these distributions parametrized by √r and 1/√r for r ∈ ℕ. For the first such distribution, we compute the moments up to the eighth degree in closed form via Poisson summation, exploiting that the hyperbolic secant is a fixed point of the −2πi-Fourier transform. As a byproduct, we obtain closed-form values for series of odd powers of the hyperbolic secant up to the ninth degree, e.\,g.\ ∑_k ∈ ℤ sech^3(πk) = (G^3+G)/√2, and reinterpret several classical Ramanujan series probabilistically. Finally, we apply our results to evaluate a contour integral, on the critical line, of the product of Dirichlet's beta, gamma, and Riemann zeta functions.
ER  -