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Roberto Tauraso

Publications and source records attributed to Roberto Tauraso.

47 records · Page 3Linked to original sources
arXiv2010

New congruences for central binomial coefficients

Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We also prove that if p>3 then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number.

math.NT↗
arXiv2009

Congruences involving alternating multiple harmonic sum

We show that for any prime prime $p\not=2$ $$\sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the l.h.s. as a combination of alternating multiple harmonic sums.

math.NT↗
arXiv2009

Congruences of alternating multiple harmonic sums

In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers and the Euler polynomials are also essential.

math.NT↗
arXiv2009

New harmonic number identities with applications

We determine the explicit formulas for the sum of products of homogeneous multiple harmonic sums $\sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{λ_j})$ when $\sum_{j=1}^r λ_j\leq 5$. We apply these identities to the study of two congruences modulo a power of a prime.

math.NT↗
arXiv2009

An elementary proof of a Rodriguez-Villegas supercongruence

We give a short proof of the following known congruence: for every odd prime $p$ $$\sum_{k=0}^{p-1}{2k\choose k}^2 16^{-k}\equiv (-1)^{p-1\over 2}\pmod{p^2}.$$ Moreover, we provide some new results connected with the above congruence.

math.NT↗
arXiv2008

The operad Lie is free

We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators, finding a recursive formula for its coefficients, and showing that the asymptotic density of the operadic generators is 1/e.

math.QA↗
arXiv2007

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period $ν_m(q)$ such that for any integers n>0 and r we have $[n+ν_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.

math.NT↗
arXiv2005

The dinner table problem: the rectangular case

$n$ people are seated randomly at a rectangular table with $\lfloor n/2\rfloor$ and $\lceil n/2\rceil$ seats along the two opposite sides for two dinners. What's the probability that neighbors at the first dinner are no more neighbors at the second one? We give an explicit formula and we show that its asymptotic behavior as $n$ goes to infinity is $e^{-2}(1+4/n)$ (it is known that it is $e^{-2}(1-4/n)$ for a round table). A more general permutation problem is also considered.

math.CO↗