The one-parameter family of polynomials $J_n(x,y)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{y+j}{j}\right)x^j$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n\ge 6$, the polynomial $J_n(x,y_0)$ is irreducible over $\mathbb{Q}$ for all but finitely many $y_0\in \mathbb{Q}$. If $n$ is odd, then with the exception of a finite set of $y_0$, the Galois group of $J_n(x,y_0)$ is $S_n$; if $n$ is even, then the exceptional set is thin.

Les recherches sur les ovales au xixe témoignent du renouveau des méthodes géométriques et illustrent la mise en concurrence de ces méthodes avec les calculs analytiques. En particulier, elles interviennent dans les relations entre l’algèbre des fonctions elliptiques et la géométrie des courbes, que les mathématiciens pensent en termes d’application ou d’interprétation d’un domaine dans l’autre. La rectification des ovales en arcs d’ellipses est obtenue dans les années 1850 par Roberts et Genocchi,...

2000 Math. Subject Classification: 33E12, 65D20, 33F05, 30E15The paper deals with analysis of several techniques and methods for the numerical evaluation of the Wright function. Even if the focus is mainly on the real arguments’ values, the methods introduced here can be used in the complex plane, too. The approaches presented in the paper include integral representations of the Wright function, its asymptotic expansions and summation of series. Because the Wright function depends on two parameters ...

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_R$ associated with the Jacobi transform through the defining relation $\widehat{S_{R}f}\left(\lambda \right)=1_{\left|\lambda \right|\le R}f̂\left(t\right)$ for a function f on ℝ is shown to be bounded from $L^p(ℝ₊,d\mu )$ into $L^p(ℝ,d\mu )+L²(ℝ,d\mu )$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p₀,1}(ℝ₊,d\mu )$ into $L^{p₀,\infty }(ℝ,d\mu )+L²(ℝ,d\mu )$. In particular ${S_{R}f\left(t\right)}_{R>0}$ converges almost everywhere towards f, for $f\in L^p(ℝ₊,d\mu )$, whenever (4α + 4)/(2α + 3) < p ≤ 2.

Zeta-generalized-Euler-constant functions, $$\gamma \left(s\right):=\sum _{k=1}^{\infty }\left(\frac{1}{k^s}-{\int }_k^{k+1}\frac{dx}{x^s}\right)$$ and $$\tilde{\gamma }\left(s\right):=\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}\left(\frac{1}{k^s}-{\int }_k^{k+1}\frac{dx}{x^s}\right)$$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $$\tilde{\gamma }$$ (1) = ln $$\frac{4}{\pi }$$ , are studied and estimated with high accuracy.

A dual transformation is discussed, by which a concurrent chart represented by one equation is transformed into an alignment chart or into a tangential contact chart. Using this transformation an alignment chart where three scales coincide and a tangential contact chart consisting of a family of circles, which represent the relation $u+v+w=0$, are constructed. In this case, a form of the addition-theorem for Weierstrass’ function involving no derivative is used.

We present an algorithm for constructing the free algebra over a given finite partial algebra in the variety determined by a finite list of equations. The algorithm succeeds whenever the desired free algebra is finite.