Congruences for certain binomial sums

Jung-Jo Lee

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 65-71
  • ISSN: 0011-4642

Abstract

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We exploit the properties of Legendre polynomials defined by the contour integral 𝐏 n ( z ) = ( 2 π i ) - 1 ∮ ( 1 - 2 t z + t 2 ) - 1 / 2 t - n - 1 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 ⩾ 5 and n = r p 2 - 1 , we have ∑ k = 0 ⌊ n / 2 ⌋ 2 k k ≡ 0 , 1 or - 1 ( mod p 2 ) , depending on the value of r ( mod 6 ) .

How to cite

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Lee, Jung-Jo. "Congruences for certain binomial sums." Czechoslovak Mathematical Journal 63.1 (2013): 65-71. <http://eudml.org/doc/252517>.

@article{Lee2013,
abstract = {We exploit the properties of Legendre polynomials defined by the contour integral $\mathbf \{P\}_n(z)=(2\pi \{\rm i\})^\{-1\} \oint (1-2tz+t^2)^\{-1/2\}t^\{-n-1\} \{\rm d\} t,$ 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 $\sum _\{k=0\}^\{\lfloor n/2\rfloor \}\{2k \atopwithdelims ()k\}\equiv 0, 1\text\{ or \}-1 \hspace\{4.44443pt\}(\@mod \; p^2)$, depending on the value of $r \hspace\{4.44443pt\}(\@mod \; 6)$.},
author = {Lee, Jung-Jo},
journal = {Czechoslovak Mathematical Journal},
keywords = {central binomial coefficient; Legendre polynomial; central binomial coefficient; Legendre polynomial; congruence},
language = {eng},
number = {1},
pages = {65-71},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Congruences for certain binomial sums},
url = {http://eudml.org/doc/252517},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Lee, Jung-Jo
TI - Congruences for certain binomial sums
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 65
EP - 71
AB - We exploit the properties of Legendre polynomials defined by the contour integral $\mathbf {P}_n(z)=(2\pi {\rm i})^{-1} \oint (1-2tz+t^2)^{-1/2}t^{-n-1} {\rm d} t,$ 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 $\sum _{k=0}^{\lfloor n/2\rfloor }{2k \atopwithdelims ()k}\equiv 0, 1\text{ or }-1 \hspace{4.44443pt}(\@mod \; p^2)$, depending on the value of $r \hspace{4.44443pt}(\@mod \; 6)$.
LA - eng
KW - central binomial coefficient; Legendre polynomial; central binomial coefficient; Legendre polynomial; congruence
UR - http://eudml.org/doc/252517
ER -

References

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  1. Callan, D., On generating functions involving the square root of a quadratic polynomial, J. Integer Seq. 10 (2007), Article 07.5.2. (2007) Zbl1138.05300MR2304410
  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. (2009) MR1542130
  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). (2007) MR2360010
  4. Mattarei, S., Asymptotics of partial sums of central binomial coefficients and Catalan numbers, arXiv:0906.4290v3. 

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