Geodesic
Congruences for certain binomial sums
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 65-71.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 $r \pmod 6$.
DOI : 10.1007/s10587-013-0004-6
Classification : 05A10, 05A19, 11A07, 11B65
Keywords: central binomial coefficient; Legendre polynomial 
@article{10_1007_s10587_013_0004_6,
     author = {Lee, Jung-Jo},
     title = {Congruences for certain binomial sums},
     journal = {Czechoslovak Mathematical Journal},
     pages = {65--71},
     publisher = {mathdoc},
     volume = {63},
     number = {1},
     year = {2013},
     doi = {10.1007/s10587-013-0004-6},
     mrnumber = {3035497},
     zbl = {1274.11052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0004-6/}
}
TY  - JOUR
AU  - Lee, Jung-Jo
TI  - Congruences for certain binomial sums
JO  - Czechoslovak Mathematical Journal
PY  - 2013
SP  - 65
EP  - 71
VL  - 63
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0004-6/
DO  - 10.1007/s10587-013-0004-6
LA  - en
ID  - 10_1007_s10587_013_0004_6
ER  - 
%0 Journal Article
%A Lee, Jung-Jo
%T Congruences for certain binomial sums
%J Czechoslovak Mathematical Journal
%D 2013
%P 65-71
%V 63
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0004-6/
%R 10.1007/s10587-013-0004-6
%G en
%F 10_1007_s10587_013_0004_6
Lee, Jung-Jo. Congruences for certain binomial sums. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 65-71. doi : 10.1007/s10587-013-0004-6. http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0004-6/

Cité par Sources :