Scaling Limits for Mixed Kernels

Let \(\mu \) and \(\nu \) be measures supported on \(\left( -1,1\right) \) with corresponding orthonormal polynomials \(\left\{ p_{n}^{\mu }\right\} \) and \( \left\{ p_{n}^{\nu }\right\} \), respectively. Define the mixed kernel

$$\begin{aligned} K_{n}^{{\mu },\nu }\left( x,y\right) =\sum _{j=0}^{n-1}p_{j}^{\mu }\left( x\right) p_{j}^{\nu }\left( y\right) . \end{aligned}$$

We establish scaling limits such as

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{\pi \sqrt{1-\xi ^{2}}\sqrt{\mu ^{\prime }\left( \xi \right) \nu ^{\prime }\left( \xi \right) }}{n}K_{n}^{\mu ,\nu }\left( \xi +\frac{a\pi \sqrt{1-\xi ^{2}}}{n},\xi +\frac{b\pi \sqrt{1-\xi ^{2}}}{n}\right) \\&\quad =S\left( \frac{\pi \left( a-b\right) }{2}\right) \cos \left( \frac{\pi \left( a-b\right) }{2}+B\left( \xi \right) \right) , \end{aligned}$$

where \(S\left( t\right) =\frac{\sin t}{t}\) is the sinc kernel, and \(B\left( \xi \right) \) depends on \({\mu },\nu \) and \(\xi \). This reduces to the classical universality limit in the bulk when \(\mu =\nu \). We deduce applications to the zero distribution of \(K_{n}^{{\mu },\nu }\), and asymptotics for its derivatives.

     

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.

     

Fractional sums and Euler-like identities

We introduce a natural definition for sums of the form

$\sum_{\nu=1}^xf(\nu)$

when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the Γ function or Euler’s little-known formula \(\sum_{\nu=1}^{-1/2}\frac{1}{\nu}=-2\ln 2\).

Many classical identities like the geometric series and the binomial theorem nicely extend to this more general setting. Sums with a fractional number of terms are closely related to special functions, in particular the Riemann and Hurwitz ζ functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like

$\begin{array}{l}\displaystyle\lim_{n\to\infty}\Biggl[e^{\frac{n}{4}(4n+1)}n^{-\frac{1}{8}-n(n+1)}(2\pi)^{-\frac{n}{2}}\prod_{k=1}^{2n}\Gamma\biggl(1+\frac{k}{2}\biggr)^{k(-1)^k}\Biggr]\\[12pt]\quad =\displaystyle\sqrt[12]{2}\exp\biggl(\frac{5}{24}-\frac{3}{2}\zeta'(-1)-\frac{7\zeta(3)}{16\pi^2}\biggr),\end{array}$

some of which seem to be new; and even for those which are known, our approach provides a new method to derive these identities and many others.

     

Inequalities for combinatorial sums

For \(k,l\in \mathbf {N}\), let

$$\begin{aligned}&P_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k-1} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }\\&\quad \text{ and }\quad Q_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }. \end{aligned}$$

We prove that the inequality

$$\begin{aligned} \frac{1}{4}\le P_{k,l} \end{aligned}$$

is valid for all natural numbers k and l. The sign of equality holds if and only if \(k=l=1\). This complements a result of Vietoris, who showed that

$$\begin{aligned} P_{k,l}<\frac{1}{2} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$

An immediate corollary is that

$$\begin{aligned} \frac{1}{4}\le P_{k,l}<\frac{1}{2} <Q_{k,l}\le \frac{3}{4} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$

The constant bounds are sharp.

     

Ramanujan-like series for $$\frac{1}{\pi }$$ involving harmonic numbers

We introduce new classes of Ramanujan-like series for \(\frac{1}{\pi }\), by devising methods for evaluating harmonic sums involving squared central binomial coefficients, such as the Ramanujan-type series

$$\begin{aligned} \sum _{n=1}^{\infty } \frac{\left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 \left( H_n^2+H_n^{(2)}\right) }{16^n (2 n-1)} = \frac{4 \pi }{3}-\frac{32 \ln ^2(2) - 32 \ln (2) + 16 }{\pi } \end{aligned}$$

introduced in this article. While the main technique used in this article is based on the evaluation of a parameter derivative of a beta-type integral, we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for \(\frac{1}{\pi }\) containing harmonic numbers.

     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\)

$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$

and \(w_y\) is the unique solution of

$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$

for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\)

$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$

The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)

$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$

(1)

     

A new Stirling series as continued fraction

We introduce the following new Stirling series

$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $

as a continued fraction, which is faster than the classical Stirling series.

     

Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent:

$$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$

(0.1)

For \(N\ge 4\) and \(\sigma \in (\frac{1}{2}, 1),\) we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).

     

Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).

     

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta <n\) and \(\tau >0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have

$$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$

Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta <2m\) and \(\tau >0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that

$$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$

for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use

$$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$

The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$

where the nonlinearity f has the critical exponential growth.

     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

Approximation of Sondow’s generalized-Euler-constant function on the interval [−1, 1]

Using the Euler-Maclaurin (Boole/Hermite) summation formula, the generalized-Euler-Sondow-constant function γ(z),

$ \gamma(z):=\sum_{k=1}^{\infty}z^{k-1}\left(\frac{1}{k}-\ln\frac{k+1}{k}\right) \qquad (-1\le z\le 1),$

where \({\gamma(-1)=\ln\frac{4}{\pi}}\) and γ(1) is the Euler-Mascheroni constant, is estimated accurately.

     

On some congruences of certain binomial sums

For any prime \(p>3,\) we prove that

$$\begin{aligned} \sum _{k=0}^{p-1}\frac{3k+1}{(-8)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-1}{p}\right) +p^3E_{p-3}\pmod {p^4}, \end{aligned}$$

where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers and \(\left( \frac{\cdot }{p}\right) \) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime \(p,\)

$$\begin{aligned} \sum _{k=0}^{\frac{p-1}{2}}\frac{6k+1}{(-512)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-2}{p}\right) \pmod {p^2} \end{aligned}$$

using WZ method.

     

Evaluation of Series Involving the Product of the Tail of $${\zeta(k)}$$ and $${\zeta(k+1)}$$ζ(k+1)

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

     

Analytic Hypoellipticity for a New Class of Sums of Squares of Vector Fields in $${\mathcal {R}}^3$$

This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(x, y) quasi-homogeneous in (x, y), consider the vector fields

$$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$

\( n_1, n_2 \ne 0\). We show that the operator

$$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$

well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).

     

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$