The authors prove meromorphic continuation in t of fundamental solutions for two nonlocal parabolic equations linked to logarithmic Laplacians by using eigenvalue asymptotics a ln(n) + O(1) and explicit eigenfunctions, expressing the solutions via zeta-like series and deriving related special-functi
Asymptotic expansions of exponentials of digamma function and identity for Bernoulli polynomials
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abstract
The asymptotic expansion of digamma function is a starting point for the derivation of approximants for harmonic sums or Euler-Mascheroni constant. It is usual to derive such approximations as values of logarithmic function, which leads to the expansion of the exponentials of digamma function. In this paper the asymptotic expansion of the function $\exp(p\psi(x+t))$ is derived and analyzed in details, especially for integer values of parameter $p$. The behavior for integer values of $p$ is proved and as a consequence a new identity for Bernoulli polynomials. The obtained formulas are used to improve know inequalities for Euler's constant and harmonic numbers.
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On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians
The authors prove meromorphic continuation in t of fundamental solutions for two nonlocal parabolic equations linked to logarithmic Laplacians by using eigenvalue asymptotics a ln(n) + O(1) and explicit eigenfunctions, expressing the solutions via zeta-like series and deriving related special-functi