Jarvis, Frazer; Verrill, Helena A. Supercongruences for the Catalan-Larcombe-French numbers. (English) Zbl 1218.11025 Ramanujan J. 22, No. 2, 171-186 (2010). The Catalan-Larcombe-French numbers \[ P_n={1\over n!}\sum_{p+q=n} {2p\choose p}{2q\choose q}{(2p)!(2q)!\over p!q!} \] has been studied in several previous papers (among others) by P. J. Larcombe and D. R. French [Congr. Numerantium 143, 33–64 (2000; Zbl 0971.05001) and ibid. 148, 65–91 (2001; Zbl 0999.05003)] and also together with A. F. Jarvis [Congr. Numerantium 161, 151–162 (2003; Zbl 1054.11016) and Indian J. Math. 47, No. 2–3, 159–181 (2005; Zbl 1114.11021)]. Some divisibility properties of \(P_n\) by primes \(p\) were proved in the last paper and some related conjectures are made here.In the present paper two of these conjectures (Conjecture 3 and 4 on p. 19) are proved. Then it is shown that the generating function for \(\{P_n\}_n\) satisfies certain second-order differential equation, which in turn is interpreted as Picard-Fuchs equation for a pencil of elliptic curves. Then developing this connection on the background of J. Stienstra and F. Beukers’ approach [Math. Ann. 271, 269–304 (1985; Zbl 0539.14006)] the “supercongruence” \(P_{mp^r}\equiv P_{mp^{r-1}}\pmod{p^r}\) is proved. Finally, the authors also show how this supercongruence and supercongruences for related numbers can be proved using A. Granville’s method [Organic mathematics. CMS Conf. Proc. 20, 253–276 (1997) (1995; Zbl 0903.11005)]. Reviewer: Štefan Porubský (Praha) Cited in 24 Documents MSC: 11B83 Special sequences and polynomials 05A10 Factorials, binomial coefficients, combinatorial functions 11B65 Binomial coefficients; factorials; \(q\)-identities 11F30 Fourier coefficients of automorphic forms 33E05 Elliptic functions and integrals Keywords:supercongruences; recurrence relation; Picard-Fuchs equation; binomial coefficient; Franel numbers; Catalan-Larcombe-French numbers Citations:Zbl 0971.05001; Zbl 0999.05003; Zbl 1054.11016; Zbl 1114.11021; Zbl 0539.14006; Zbl 0903.11005 Software:OEIS PDFBibTeX XMLCite \textit{F. Jarvis} and \textit{H. A. Verrill}, Ramanujan J. 22, No. 2, 171--186 (2010; Zbl 1218.11025) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2. Catalan-Larcombe-French sequence. Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q. References: [1] Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics, vol. 41. Springer, New York (1976) · Zbl 0332.10017 [2] Beukers, F.: Another congruence for the Apéry numbers. J. Number Theory 25, 201–210 (1987) · Zbl 0614.10011 [3] Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley–Interscience, New York (1987) · Zbl 0611.10001 [4] Catalan, E.: Sur les nombres de Segner. Rend. Circ. Mat. Palermo 1, 190–201 (1887) · JFM 19.0170.03 [5] Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979) · Zbl 0424.20010 [6] Cusick, T.W.: Recurrences for sums of powers of binomial coefficients. J. Comb. Theory, Ser. A 52, 77–83 (1989) · Zbl 0682.05006 [7] Fang, L., Hoffman, J.W., Linowitz, B., Rupinski, A., Verrill, H.A.: Modular forms on noncongruence subgroups and Atkin–Swinnerton–Dyer relations. arXiv:0805.2144 [math.NT] · Zbl 1225.11049 [8] Fine, N.J.: Basic Hypergeometric Series and Applications. With a foreword by George E. Andrews. Mathematical Surveys and Monographs, vol. 27. Am. Math. Soc., Providence (1988). xvi+124 pp. ISBN:0-8218-1524-5 [9] Granville, A.: Arithmetic properties of binomial coefficients I. Binomial coefficients modulo prime powers. In: Organic Mathematics, Burnaby, BC, 1995. CMS Conf. Proc., vol. 20, pp. 253–276. Am. Math. Soc., Providence (1997) · Zbl 0903.11005 [10] Jarvis, A.F., Larcombe, P., French, D.: Applications of the A.G.M. of Gauss: some new properties of the Catalan–Larcombe–French sequence. Congr. Numer. 161, 151–162 (2003). Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing · Zbl 1054.11016 [11] Jarvis, A.F., Larcombe, P., French, D.: On small prime divisibility of the Catalan–Larcombe–French numbers. Indian J. Math. 47, 159–181 (2005) · Zbl 1114.11021 [12] Larcombe, P., French, D.: On the ’other’ Catalan numbers: a historical formulation re-examined. Congr. Numer. 143, 33–64 (2000) · Zbl 0971.05001 [13] Martin, Y.: Multiplicative {\(\eta\)}-quotients. Trans. Am. Math. Soc. 348, 4825–4856 (1996) · Zbl 0872.11026 [14] Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, vol. 11. Princeton University Press, Princeton (1994), xiv+271 pp. Reprint of the 1971 original. ISBN:0-691-08092-5. · Zbl 0872.11023 [15] Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, published electronically at www.research.att.com/\(\sim\)njas/sequences/ [16] Stienstra, J., Beukers, F.: On the Picard–Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math. Ann. 271, 271–304 (1985) · Zbl 0555.14006 [17] Verrill, H.A.: Picard–Fuchs Equations of Some Families of Elliptic Curves, Proceedings on Moonshine and Related Topics, Montréal, Québec, 1999. CRM Proc. Lecture Notes, vol. 30, pp. 253–268. Am. Math. Soc., Providence (2001) · Zbl 1082.14503 [18] Verrill, H.A.: Some Congruences Related to modular forms. Max Planck Institut für Mathematik preprint 26 (1999) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.