Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum

Abstract

This is the translation of Euler's Latin textbook Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum (second volume) into English.

arXiv commentContains 18 Chapters, each of them can be understood as a single paper. All translations can also be found at https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

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• The $q$-tangent and $q$-secant numbers via continued fractions
• Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to ${\pi }^{-1}$
• Expansions of generalized Euler's constants into the series of polynomials in ${\pi }^{-2}$ and into the formal enveloping series with rational coefficients only
• Error bounds for the asymptotic expansion of the Hurwitz zeta function
• Central limit theorem for descents in conjugacy classes of $S_n$
• An asymptotic distribution theory for Eulerian recurrences with applications
• Asymptotics of Certain Sums Required in Loop Regularisation
• Faulhaber polynomials and reciprocal Bernoulli polynomials
• The Primitive Eulerian polynomial
• On the Möbius function of permutations under the pattern containment order