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The p-adic valuations of the coefficients in Yang's expansion of (1+1/x)^x

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Yang's coefficients b_n, defined by (1+1/x)^x = e(1 - sum_{n>=1} b_n (x+1)^{-n}), appear in refinements of Carleman's inequality, and the literature on them concerns their analytic properties. We study their p-adic valuations for an odd prime p. From the identity 1 - sum_n b_n u^n = exp(-sum_{k>=1} u^k/(k(k+1))) one gets v_p(b_n) >= -M(n), where M(n) is the optimum of a knapsack problem on the compositions of n. We compute M(n) in closed form and show that, for n >= 2, the optimum is attained by a unique composition exactly when n = (p-1)m with m in an explicit set U_p defined by a digit condition. For these n the bound is an equality: v_p(b_n) = -(m + v_p(m!)). The set U_p has density 1/(p-1), so the valuation of b_n is determined on a set N_p of indices of density 1/(p-1)^2. The set N_p contains the indices p^K - 1, at which b_{p^K-1} is the unique coefficient of minimal valuation among b_1, ..., b_{p^K-1}. The same statements hold for the coefficients of e(1-u)^{1/u}, and the minimality statement for those of e(1-u)^{1/u-1}. A script checks all statements numerically in finite ranges.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. T. Carleman, Sur les fonctions quasi-analytiques, in: Conférences faites au cinquième congrès des mathématiciens scandinaves, Helsingfors, 1923, 181--196.
  2. L. Comtet, Advanced Combinatorics , Reidel, Dordrecht, 1974.
  3. D. M. Davis, Coefficients in powers of the log series, arXiv:1001.3068 (2010).arXiv
  4. S. R. Finch, Carleman's inequality, note dated 17 December 2013, https://oeis.org/A219245/a219245.pdf.link
  5. Y. Hu, C. Mortici, On the coefficients of an expansion of (1+1/x)x(1+1/x)^x related to Carleman's inequality, arXiv:1401.2236 (2014).arXiv
  6. C. Krattenthaler, T. Rivoal, Arithmetic properties of the Taylor coefficients of differentially algebraic power series, arXiv:2502.09259 (2025).arXiv
  7. The On-Line Encyclopedia of Integer Sequences, sequences A055505, A055535, A249276 and A249277, https://oeis.org.link
  8. X. Yang, Approximations for constant ee and their applications, J. Math. Anal. Appl. 262 (2001), 651--659.

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026