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Keywords:
central binomial coefficient; Legendre polynomial
Summary:
We exploit the properties of Legendre polynomials defined by the contour integral $\bold P_n(z)=(2\pi {\rm i})^{-1} \oint (1-2tz+t^2)^{-1/2}t^{-n-1} {\rm d} t,$ where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer $r$, a prime $p \geqslant 5$ and $n=rp^2-1$, we have $\sum _{k=0}^{\lfloor n/2\rfloor }{2k \choose k}\equiv 0, 1\text { or }-1 \pmod {p^2}$, depending on the value of $r \pmod 6$.
References:
[1] Callan, D.: On generating functions involving the square root of a quadratic polynomial. J. Integer Seq. 10 (2007), Article 07.5.2. MR 2304410 | Zbl 1138.05300
[2] Callan, D., Chapman, R.: Divisibility of a central binomial sum (Problems and Solutions 11292&11307 [2007, 451&640]). American Mathematical Monthly 116 (2009), 468-470.
[3] Gradshteyn, I. S., Ryzhik, I. M.: Table of Integrals, Series, and Products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger 7th ed. Elsevier/Academic Press, Amsterdam (2007). MR 2360010
[4] Mattarei, S.: Asymptotics of partial sums of central binomial coefficients and Catalan numbers. arXiv:0906.4290v3.
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