Hao Pan

Nanjing University of Finance and Economics

*This is an algorithmically generated author record based on publicly available information.

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Nanjing University of Finance and Economics

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Research Subjects

Mathematical Theory

Research Interests

Hao Pan has a total of 49 co-authors. Here are the top co-authors by number of shared publications:

Zhi-Wei Sun
School of Mathematics, Nanjing University, Nanjing 210093, China
23 shared publications
Hongze Li
School of Mathematical Sciences , 12474 Shanghai Jiao Tong University , Shanghai , P. R. China
12 shared publications
Li-Xia Dai
School of Mathematical Sciences, Nanjing Normal University, Nanjing, People’s Republic of China
9 shared publications
Hui-Qin Cao
Department of Applied Mathematics, Nanjing Audit University, Nanjing, People’s Republic of China
8 shared publications
Yong Zhang
School of Mathematics and Physics Nanjing Institute of Technology 211167 Nanjing China
7 shared publications
Chen Wang
Department of Applied Mathematics, Nanjing Forestry University, Nanjing, People’s Republic of China
4 shared publications

Latest Publications

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- 17 September 2026
- Published by Springer Nature in Indian Journal of Pure and Applied Mathematics
Laguerre polynomials and Appell sequences
by Min-Jing Sun, Hao Pan
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- 17 September 2026
- Published by Springer Nature in Indian Journal of Pure and Applied Mathematics
Laguerre polynomials and Appell sequences
by Min-Jing Sun, Hao Pan
Open Access
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- 19 March 2026
- Published by MDPI AG in Mathematics
Analytic Aspects of Weighted Delannoy Numbers
by Lei Chen, Yu-Yang Zhang, Hao Pan
The weighted Delannoy number $D^{(a, b, c)} (h, k)$ is a weighted generalization of the Delannoy number. We show that all zeros of the polynomial $d_{n}^{(a, b, c)} (x) : = \sum_{k = 0}^{n} D^{(a, b, c)} (n - k, k) x^{k}$ are real, distinct and lying in the open interval $(- \frac{{(\sqrt{a b + c} + \sqrt{c})}^{2}}{b^{2}}, - \frac{{(\sqrt{a b + c} - \sqrt{c})}^{2}}{b^{2}})$ . Using this result, we also discuss the asymptotic normality of $D^{(a, b, c)} (n - k, k)$ .
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- 19 March 2026
- Published by MDPI AG in Mathematics
Open Access
Analytic Aspects of Weighted Delannoy Numbers
by Lei Chen, Yu-Yang Zhang, Hao Pan
The weighted Delannoy number $D^{(a, b, c)} (h, k)$ is a weighted generalization of the Delannoy number. We show that all zeros of the polynomial $d_{n}^{(a, b, c)} (x) : = \sum_{k = 0}^{n} D^{(a, b, c)} (n - k, k) x^{k}$ are real, distinct and lying in the open interval $(- \frac{{(\sqrt{a b + c} + \sqrt{c})}^{2}}{b^{2}}, - \frac{{(\sqrt{a b + c} - \sqrt{c})}^{2}}{b^{2}})$ . Using this result, we also discuss the asymptotic normality of $D^{(a, b, c)} (n - k, k)$ .
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- 1 January 2026
- Published by American Institute of Mathematical Sciences (AIMS) in Frontiers in Combinatorics and Number Theory
Supercongruences for central trinomial coefficients
by Hao Pan, Zhi-Wei Sun
For each $ n = 0, 1, 2, \ldots $, the central trinomial coefficient $ T_n $ is the coefficient of $ x^n $ in the expansion of $ (x^2+x+1)^n $. Let $ p>3 $ be a prime, and let $ n $ be any positive integer. In 2016, the second author conjectured that the quotient $ (T_{pn}-T_n)/(pn)^2 $ is always a $ p $-adic integer. In this paper, we confirm this conjecture, and further prove that $ \frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p, $ where $ (\frac p3) $ is the Legendre symbol and $ B_{p-2}(x) $ is the Bernoulli polynomial of degree $ p-2 $.
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- 1 January 2026
- Published by American Institute of Mathematical Sciences (AIMS) in Frontiers in Combinatorics and Number Theory
Supercongruences for central trinomial coefficients
by Hao Pan, Zhi-Wei Sun
For each $ n = 0, 1, 2, \ldots $, the central trinomial coefficient $ T_n $ is the coefficient of $ x^n $ in the expansion of $ (x^2+x+1)^n $. Let $ p>3 $ be a prime, and let $ n $ be any positive integer. In 2016, the second author conjectured that the quotient $ (T_{pn}-T_n)/(pn)^2 $ is always a $ p $-adic integer. In this paper, we confirm this conjecture, and further prove that $ \frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p, $ where $ (\frac p3) $ is the Legendre symbol and $ B_{p-2}(x) $ is the Bernoulli polynomial of degree $ p-2 $.
Open Access
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- 13 November 2025
- Published by Institute of Mathematics, Polish Academy of Sciences in Acta Arithmetica
On finite field analogues of determinants involving the Beta function
by Hai-Liang Wu, Li-Yuan Wang, Hao Pan
Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, we investigate certain cyclotomic matrices concerning Jacobi sums over finite fields, which can be viewed as finite field analogues of certain matrices involving the Beta function. For example, let q>1 be a prime power and let χ be a generator of the group of all multiplicative characters of Fq. Then we prove that det[Jq(χi,χj)]1≤i,j≤q−2=(q−1)q−3, where Jq(χi,χj) is the Jacobi sum over Fq. This is a finite analogue of det[B(i,j)]1≤i,j≤n=(−1)n(n−1)/2∏r=0n−1(r!)3(n+r)!, where B is the Beta function. Also, if q=p≥5 is an odd prime, then we show that det[Jp(χ2i,χ2j)]1≤i,j≤(p−3)/2=1+(−1)(p+1)/2p4(p−12)(p−5)/2.
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- 13 November 2025
- Published by Institute of Mathematics, Polish Academy of Sciences in Acta Arithmetica
Open Access
On finite field analogues of determinants involving the Beta function
by Hai-Liang Wu, Li-Yuan Wang, Hao Pan
Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, we investigate certain cyclotomic matrices concerning Jacobi sums over finite fields, which can be viewed as finite field analogues of certain matrices involving the Beta function. For example, let q>1 be a prime power and let χ be a generator of the group of all multiplicative characters of Fq. Then we prove that det[Jq(χi,χj)]1≤i,j≤q−2=(q−1)q−3, where Jq(χi,χj) is the Jacobi sum over Fq. This is a finite analogue of det[B(i,j)]1≤i,j≤n=(−1)n(n−1)/2∏r=0n−1(r!)3(n+r)!, where B is the Beta function. Also, if q=p≥5 is an odd prime, then we show that det[Jp(χ2i,χ2j)]1≤i,j≤(p−3)/2=1+(−1)(p+1)/2p4(p−12)(p−5)/2.
Open Access
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- 27 April 2025
- Published by MDPI AG in Mathematics
Analytic Aspects of the Generalized Second-Order Central Trinomial Coefficients
by Yu-Yang Zhang, Hao Pan, Lei Chen
The generalized second-order central trinomial coefficients $T_{n}^{(2)} (b, c)$ are a special case of the generalized r-order central trinomial coefficients, corresponding to $r = 2$ . We show that all zeros of the polynomial ${T^{(2)}}_{n} (x) = \sum_{k = 0}^{⌊ n / 2 ⌋} T^{(2)} (n, k) x^{k}$ are real. And zeros of ${T^{(2)}}_{n - 1} (x)$ interlace those of ${T^{(2)}}_{n - 2} (x)$ , as well as those of ${T^{(2)}}_{n} (x)$ . Using this result, we also discuss the asymptotic normality of $T^{(2)} (n, k)$ .
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- 27 April 2025
- Published by MDPI AG in Mathematics
Open Access
Analytic Aspects of the Generalized Second-Order Central Trinomial Coefficients
by Yu-Yang Zhang, Hao Pan, Lei Chen
The generalized second-order central trinomial coefficients $T_{n}^{(2)} (b, c)$ are a special case of the generalized r-order central trinomial coefficients, corresponding to $r = 2$ . We show that all zeros of the polynomial ${T^{(2)}}_{n} (x) = \sum_{k = 0}^{⌊ n / 2 ⌋} T^{(2)} (n, k) x^{k}$ are real. And zeros of ${T^{(2)}}_{n - 1} (x)$ interlace those of ${T^{(2)}}_{n - 2} (x)$ , as well as those of ${T^{(2)}}_{n} (x)$ . Using this result, we also discuss the asymptotic normality of $T^{(2)} (n, k)$ .
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- 30 December 2024
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
Ergodic recurrence and bounded gaps between primes
by Hao Pan
Let $(\mathcal {X},\mathscr {B}_\mathcal {X},\mu ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in \mathscr {B}_\mathcal {X}$ with $\mu (A)>0$ and $\epsilon >0$ . For any $m\geq 1$ , there exist infinitely many primes $p_0,p_1,\ldots ,p_m$ with $p_0>\cdots >p_m$ such that $\begin{equation*} \mu (A\cap T^{-(p_i-1)}A)\geq \mu (A)^2-\epsilon \end{equation*}$ for each $0\leq i\leq m$ and $\begin{equation*} p_m-p_0>C_{m,A,\epsilon }, \end{equation*}$ where $C_{m,A,\epsilon }>0$ is a constant only depending on $m$ , $A$ and $\epsilon$ .
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- 30 December 2024
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
Ergodic recurrence and bounded gaps between primes
by Hao Pan
Let $(\mathcal {X},\mathscr {B}_\mathcal {X},\mu ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in \mathscr {B}_\mathcal {X}$ with $\mu (A)>0$ and $\epsilon >0$ . For any $m\geq 1$ , there exist infinitely many primes $p_0,p_1,\ldots ,p_m$ with $p_0>\cdots >p_m$ such that $\begin{equation*} \mu (A\cap T^{-(p_i-1)}A)\geq \mu (A)^2-\epsilon \end{equation*}$ for each $0\leq i\leq m$ and $\begin{equation*} p_m-p_0>C_{m,A,\epsilon }, \end{equation*}$ where $C_{m,A,\epsilon }>0$ is a constant only depending on $m$ , $A$ and $\epsilon$ .
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Cited by 1
- 24 December 2024
- Published by Springer Nature in Periodica Mathematica Hungarica
Solutions to some sign change problems on functions involving sums of divisors
by Yuchen Ding, Hao Pan, Yu-Chen Sun
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- 24 December 2024
- Published by Springer Nature in Periodica Mathematica Hungarica
Solutions to some sign change problems on functions involving sums of divisors
by Yuchen Ding, Hao Pan, Yu-Chen Sun
Cited by 1
Open Access
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- 23 September 2024
- Published by University of Calgary in Contributions to Discrete Mathematics
A new trigonometric identity with applications
by Zhi-Wei Sun, Hao Pan
In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$, we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}{2},$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$, we obtain the identity $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.
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- 23 September 2024
- Published by University of Calgary in Contributions to Discrete Mathematics
Open Access
A new trigonometric identity with applications
by Zhi-Wei Sun, Hao Pan
In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$, we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}{2},$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$, we obtain the identity $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.
Open Access
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- 6 March 2024
- Published by arXiv in arXiv
The restricted sumsets in finite abelian groups
by Shanshan Du, Hao Pan
Suppose that $k\geq 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G|>1$. Then the cardinality of the restricted sumset $$ k^\wedge A:=\{a_1+\cdots+a_k:\,a_1,\ldots,a_k\in A,\ a_i\neq a_j\text{ for }i\neq j\} $$ is at least $$ \min\{p(G), k|A|-k^2+1\}, $$ where $p(G)$ denotes the least prime divisor of $|G|$.
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- 6 March 2024
- Published by arXiv in arXiv
Open Access
The restricted sumsets in finite abelian groups
by Shanshan Du, Hao Pan
Suppose that $k\geq 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G|>1$. Then the cardinality of the restricted sumset $$ k^\wedge A:=\{a_1+\cdots+a_k:\,a_1,\ldots,a_k\in A,\ a_i\neq a_j\text{ for }i\neq j\} $$ is at least $$ \min\{p(G), k|A|-k^2+1\}, $$ where $p(G)$ denotes the least prime divisor of $|G|$.
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- 22 January 2024
- Published by Springer Nature in Bulletin of the Malaysian Mathematical Sciences Society
Symmetric Identities Involving Discrete Appell Sequences
by Lei Wu, Hao Pan
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- 22 January 2024
- Published by Springer Nature in Bulletin of the Malaysian Mathematical Sciences Society
Symmetric Identities Involving Discrete Appell Sequences
by Lei Wu, Hao Pan

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