zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant. (English) Zbl 1113.11017

The aim of present paper is to study the generalized-Euler-constant function γ(z), defined for |z|1 by γ(z)= n=1 z n-1 (1 n-logn+1 n) and extended analytically to -[1,) by the integral representation γ(z)= 0 1 0 1 1-x (1-xyz)(-logxy)dxdy. This function has some very important constants in heart; γ(1)=γ=0·577 is the “Euler’s constant”, γ(-1)=log4 π is the “alternating Euler constant”, γ(1 2)=2log2 σ with

σ=123= n=1 n 1/2 n =1·661

is one of “Somos’s quadratic recurrence constant” [N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (published electronically), Sequence A112302 (2003; Zbl 1044.11108)].

The authors give various relations between this function and some well known special functions such as the Riemann zeta-function ζ(s)= n=1 n -s and the polylogarithm function Li k (z)= n=1 z n n -k , which lead to some interesting formulas and expansions for the above mentioned constants. They define

σ t =12 t3 t t= n=1 n 1/t n

for t>1 as a generalization of Somos’s quadratic recurrence constant, and they prove γ(1 t)=tlogt (t-1)σ t t-1 . They calculate γ(z) and γ ' (z) at roots of unity, which end in evaluating several related series and infinite products. Calculation of γ ' (z) is based on the Kinkelin-Bendersky hyperfactorial K function, the Weierstrass products for the gamma and Barnes G functions, and Jonquiere’s relation for the polylogarithm. These methods allow them to evaluate some double integrals, and especially considering the “Glaisher-Kinkelin constant” A=lim n 1 1 2 2 n n n -n 2 +n 2-1 12 e n 2 4 =1·282, they show that

γ ' (-1)=log2 11/6 A 6 π 3/2 eandγ '' (-1)=log2 10/3 A 24 π 4 e 13/4 -7ζ(3) 2π 2 ·

The paper contains some delicate computations and methods of analysis.


MSC:
11B99Sequences and sets
33B99Elementary classical functions
11Y60Evaluation of constants