Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders

In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as $x\to \infty$,$\frac{\mathrm{\Gamma }(x+p)\mathrm{\Gamma }(x+q)}{\mathrm{\Gamma }{(x+(p+q)/2)}^2}=\exp ?[\sum _{k=1}^n\frac{B_{2k}\left(s\right)?B_{2k}(1/2)}{k(2k?1){(x+r_0)}^{2k?1}}+R_n(x;p,q)]$where $p,q\in \mathbb{R}$ with $w=|p?q|\ne 0$ and $s=(1?w)/2$, $r_0=(p+q?1)/2$, $B_{2n+1}\left(s\right)$ are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function $x?{(?1)}^n{R}_n(x;p,q)$ is completely monotonic on $(?r_0,\infty )$ if $|p?q|<1$, which yields a sharp upper bound for $\left|R_n(x;p,q)\right|$. This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when $q\to p$.     

Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X??$, for a smooth scheme $X\in {\mathrm{Sm}}_k$ over an infinite perfect field k, is computed.The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X)/(({\mathbb{A}}^l?0)\times X)$, $X\in {\mathrm{Sm}}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety X over k and an open subscheme $U?X$.The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.