We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|\beta|\leq1/2$. For the spectral scale $n+(\alpha+\beta+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.
For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szeg\H{o} conjecture, recorded in the 1975 fourth edition of his classic monograph Orthogonal Polynomials, and strengthens it by removing the ordering assumption on the parameters. The proof transforms the Jacobi equation to angular form and compares Pr\"ufer amplitudes at equal phase. Combined with a reduction of de Oliveira Filho and a separate quadratic-transformation argument for the boundary case, the result also settles a question concerning the Lov\'asz theta number of spherical distance graphs in every dimension at least four.
We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show that, for $f$ in a nonempty Zariski open subset of the parameter space, $\mu_g=\mu_f$ implies $g=f$, with no hypothesis relating the leading terms; in dimension two this holds across all degrees, with $g$ an iterate of~$f$. We also establish finiteness results for maps with a prescribed equilibrium measure under essentially necessary hypotheses. Our approach rests on the stable manifold structure near the hyperplane at infinity. Equality of equilibrium measures yields a Zariski-dense family of common local stable-manifold germs, whose intrinsic first-order jets lead to a divisibility incidence problem on a Grassmannian. We resolve this problem by analyzing the power-map fiber and applying a generic stabilizer argument. In dimension two, applications include a characterization in terms of preperiodic sets, a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.
We give an all-level proof of the four-point $SU(2)$ AGT correspondence with four fundamental hypermultiplets at generic $\beta=-\epsilon_1/\epsilon_2$, by proving an all-level factorization formula for Selberg averages of generalized Jack polynomials. Taking a coefficientwise Jack limit of the generalized Macdonald Pieri rule, we obtain the required one-box matrix elements in the strict Cauchy dual basis. A rational corner-function identity then evaluates the sum over all parent double partitions, while an explicit total derivative of the Selberg kernel yields a triangular recursion on the Dotsenko-Fateev charge balance hyperplane. The unique solution of this recursion is the generalized Kadell formula previously verified only through finite level. Combining this result with the generalized Cauchy identity identifies each double partition term with the corresponding Nekrasov fixed-point contribution.
For positive integers $n$ and $M$, Sundaram defined $ F_{n,M}=\sum_{\substack{d\mid n\\d\le M}} Lie_{n/d}[p_d] $ and conjectured that every coefficient in its Schur expansion is a nonnegative integer. We prove the conjecture. The power-sum expansion isolates the contribution of the identity class, and the proof divides Young diagrams at the intrinsic majority threshold. If the first row or first column contains more than half of the boxes, the Bernstein creation formula separates each rectangular-cycle character value into a hook constant and a remainder supported only on short cycles. The restrictions of the trivial and sign characters to the cyclic subgroup evaluate the constants exactly, whereas a uniform binomial contraction controls every distance from the boundary at once. If neither a row nor a column has a majority, Swanson's opposite-hook estimate gives a Catalan-scale lower bound for the dimension. The Fomin--Lulov rectangular-character estimate, together with uniform bounds for the truncated M\"obius weights, then shows that the identity class dominates. Sundaram's plethystic identity consequently implies that $\prod_{r=1}^M(1-p_r)^{-1}$ is Schur-positive for every fixed $M$. We also record a degree-$12$ counterexample to a separate congruence-class product conjecture attributed to Richard Stanley.
UDC 517.587, 512.643
We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambda>0.$ Here, the point $a$ is chosen to guarantee that the spectrum of Meixner polynomials does not intersect the interval $(a, a+1).$ This inner product and its generalization already exist in the literature for various other measures. We consider a specific measure. For the resulting orthonormal Sobolev-type polynomials, we obtain a five-term recurrence relation. The associated pentadiagonal matrix is analyzed in relation to the Jacobi pencil matrix. The behavior of zeros with respect to the parameters $a$ and $\lambda$ is investigated and the Plancherel–Rotach-type asymptotic behavior is obtained.
N. Neha, A. Swaminathan· Ukrains'kyi Matematychnyi Zh...· 0 citations
Let (a_k) be the positive coefficient sequence of the normalized Riemann xi-function, and let D_{r,k} denote its consecutive Toeplitz minors. The Riemann Hypothesis is equivalent to (a_k) being a Polya frequency sequence of infinite order, and hence to nonnegativity of all Toeplitz minors. We prove that D_{r,k}>0 for every r>= 2 and k>= 10^18 r^3. This gives an explicit cubic tail scale uniform in r, in contrast with Katkova's fixed-order asymptotic positivity. The proof does not use numerically verified zeros of the Riemann zeta-function. It combines a certified complex saddle-point analysis of the moment transform, an exact q-Pascal dilation semigroup controlling every degree simultaneously, and a weighted Banach-algebra majorant for the nonlinear remainder, closed by an inertia-preservation argument. All analytic constants are certified with directed rounding in Arb ball arithmetic, and all algebraic identities are verified in exact rational arithmetic. The ancillary files reproduce every certificate. The result concerns only the tail regime k much larger than r^3 and makes no progress on the Riemann Hypothesis, which concerns the complementary region.
Wojciech Michalowski· 0 citations
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