On a sum involving Fourier coefficients of cusp forms

We improve the existing upper bound for the quantity |∑ _n⩽x a(n ²)|, where a(n ²) is the n ²th Hecke eigenvalue of a normalized holomorphic cusp form (Hecke eigenform) of the full modular group SL(2, ℤ), whenever the weight of the original holomorphic cusp form (Hecke eigenform) lies in a certain range. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 565–583, October–December, 2006.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

On a sum involving the Mangoldt function

Periodica Mathematica Hungarica - Let $$Lambda (n)$$ be the von Mangoldt function, and let [t] be the integral part of real number t. In this note we prove that the asymptotic formula...     

On the calculation of generalized binomial coefficients

The generalized binomial coefficients (_κ^λ) are defined by

$\text{Cλ(I}{}_{\mathrm{m}}\text{+ R)}\text{Cλ(I}{}_{\mathrm{m}}\text{)}\text{=}\text{Σ}\text{k=0}\text{l}\text{Σ}\text{κ}\text{(}\text{λ}\text{κ}\text{)}\text{C}{}_{\mathrm{\kappa }}\text{(R)}\text{C}{}_{\mathrm{\kappa }}\text{(I}{}_{\mathrm{m}}\text{)}$

, where the C_k(R) are the zonal polynomials of the m × m matrix R. In this paper some simple expressions are derived which allow straightforward calculation of a large number of these coefficients.     

On a sum involving the Euler totient function

In this short note, we prove that $\frac{4}{{\pi }^2}xlogx+O\left(x\right)?\sum _{n?x}\phi \left(\left[\frac{x}{n}\right]\right)?\left(\frac{1}{3}+\frac{4}{{\pi }^2}\right)xlogx+O\left(x\right),$ for $x\to \infty$, where $\phi \left(n\right)$ is the Euler totient function and $\left[t\right]$ is the integral part of real $t$. This improves recent results of Bordellès–Heyman–Shparlinski and of Dai–Pan.     

On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $$p^2$$

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.

     

On theq-log-concavity of Gaussian binomial coefficients

We give a combinatorial proof that is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l withab andlk. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients , which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775).     

Determinants, sums involving binomial coefficients, and moment sequences

In the present paper we investigate some problems connected with the positive definiteness of the sequences $j \to e^{j^\alpha } (j \in N_0 )$ and $j \to e^{ - \left| j \right|^\alpha } (j \in Z)$ , whereα≥0. For this we need and prove some results about certain determinants and finite sums that might be of independent interest.     

On the number of divisors of binomial coefficients

This paper deals with the problems of the upper and lower orders of growth of the ratios of the divisor functions of “adjacent” binomial coefficients, i.e., of the numbers of combinations of the form C _n ^k and C _n ^k+1 or C _n ^k and C _n+1 ^k . The suprema and infima of the corresponding ratios are obtained.     

关于广义三周期Fibonacci序列的二项式系数和的恒等式

首先构造了广义三周期Fibonacci序列的通项公式,然后在一定限制条件下,利用矩阵方法给出了关于广义三周期Fibonacci序列和广义三周期Lucas序列的一些二项式系数和的恒等式.