Treffer: Applications of the A. G. M. of Gauss: some new properties of the Catalan-Larcombe-French sequence

Title:
Applications of the A. G. M. of Gauss: some new properties of the Catalan-Larcombe-French sequence
Authors:
Jarvis, A. Frazer, Larcombe, Peter J., French, David R.
Publisher Information:
Combinatorial Press, Winnipeg, MB
Subject Terms:
elliptic integral of the first kind, generating series, Special sequences and polynomials, Elliptic integrals as hypergeometric functions, Factorials, binomial coefficients, combinatorial functions, Combinatorial inequalities, hypergeometric series, Catalan-Larcombe-French sequence, arithmetic-geometric mean
Document Type:
Fachzeitschrift Article
File Description:
application/xml
Access URL:
https://zbmath.org/2076936
Accession Number:
edsair.c2b0b933574d..ab317e7c756297755d76fd0c7f4e82b0
Database:
OpenAIRE

Weitere Informationen

The Catalan-Larcombe-French sequence \(\{P_s\}_{s=0}^\infty=\{1,8,80,896,10816,\dots\}\) can be defined through the expansion of the complete elliptic integral (of the first kind) \[ \int_0^{\pi/2} (1-c^2\sin^2\phi)^{-1/2}d\phi=(\pi/2)\sum_{s=0}^\infty P_s[(1-b(c))/16]^s \] using the substitution \(\theta(\phi)=\tan^{-1}[\sqrt{b}\tan\phi]\) with \(b^2(c)=1-c^2\) [ibid. 143, 33--64 (2000; Zbl 0971.05001); 148, 65--91 (2001; Zbl 0999.05003); 156, 17--25 (2002; Zbl 1026.05002); Util. Math. 60, 67--77 (2001; Zbl 1011.05004)]. The authors prove a new generating function for this sequence, then a new hypergeometric form of \(P_s\) and show how using the arithmetic-geometric mean one can produce certain initial segments of \(\{P_s\}_{s=0}^\infty\).