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Hao Pan

Publications and source records attributed to Hao Pan.

159 records · Page 9Linked to original sources
arXiv2006

A sharp result on m-covers

Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,k} such that the fractional part of sum_{s in I}m_s/n_s is theta, then there are at least 2^m such subsets of {1,...,k}. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to m-covers of the integral ring of any algebraic number field with a power integral basis.

math.NT↗
arXiv2006

On q-Euler numbers, q-Salie numbers and q-Carlitz numbers

Let $(a;q)_n=\prod_{0\le k<n}(1-aq^k)$ for n=0,1,2,.... Define q-Euler numbers $E_n(q)$, q-Salié numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows: $$\sum_{n=0}^{\infty}E_n(q)\frac{x^n}{(q,q)_n} =1/\sum_{n=0}^{\infty}\frac{q^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$ $$\sum_{n=0}^{\infty}S_n(q)\frac{x^n}{(q;q)_n} =\sum_{n=0}^{\infty}\frac{q^{n(n-1)}x^{2n}}{(q;q)_{2n}} /\sum_{n=0}^{\infty}\frac{(-1)^nq^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$ $$\sum_{n=0}^{\infty}C_n(q)\frac{x^n}{(q;q)_n} =\sum_{n=0}^{\infty}\frac{q^{n(n-1)}x^{2n+1}}{(q;q)_{2n+1}} /\sum_{n=0}^{\infty}\frac{(-1)^nq^{n(2n+1)}x^{2n+1}}{(q;q)_{2n+1}}.$$ We show that $$E_{2n}(q)-E_{2n+2^{s}t}(q)=[2^s]_{q^t} (mod (1+q)[2^s]_{q^t})$$ for any nonnegative integers n,s,t with t odd, where $[k]_q=(1-q^k)/(1-q)$; this is a q-analogue of Stern's congruence $E_{2n+2^s}=E_{2n}+2^s (mod 2^{s+1})$. We also prove that $(-q;q)_n=\prod_{0<k\le n}(1+q^k)$ divides $S_{2n}(q)$ and the numerator of $C_{2n}(q)$; this extends Carlitz's result that $2^n$ divides the Salié number $S_{2n}$ and the numerator of the Carlitz number $C_{2n}$. Our result on q-Salié numbers implies a conjecture of Guo and Zeng.

math.CO↗
arXiv2006

q-Analogue of Wilson's theorem

We give q-analogues of Wilson's theorem for the primes congruent 1 and 3 modulo 4 respectively. And q-analogues of two congruences due to Mordell and Chowla are also established.

math.NT↗
arXiv2006

On consecutive happy numbers

Let e>=1 and b>=2 be integers. For a positive integer n=\sum_{j=0}^ka_jb^j with 0<=a_j =0, where T_{e,b}^r is the r-th iteration of T_{e,b}. In this paper, we prove that there exist arbitrarily long sequences of consecutive (e,b)-happy numbers provided that e-1 is not divisible by p-1 for any prime divisor p of b-1.

math.NT↗
arXiv2006

A remark on Zoloterav's theorem

Let n>=3 be an odd integer. For any integer a prime to n, define the permutation gamma_{a,n} of {1,...,(n-1)/2} by gamma_{a,n}(x)=n-\dec{ax}_n if {ax}_n>=(n+1)/2, and {ax}_n if {ax}_n<=(n-1)/2, where {x}_n denotes the least nonnegative residue of x modulo n. In this note, we show that the sign of gamma_{a,n} coincides with the Jacobi symbol (a/n) if n=1 mod 4, and 1 if n=3 mod 4.

math.NT↗
arXiv2005

A q-analogue of Lehmer's congruence

We establish the q-analogue of a classical congruence of Lehmer. Also, the q-analogues of two congruences of Morley and Granville are given.

math.NT↗
arXiv2004

New identities involving Bernoulli and Euler polynomials. II

We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003), 709--718].

math.NT↗