Esther M. García-Caballero

Non-standard orthogonality for the little q-Laguerre polynomials

Abstract Little q -Laguerre polynomials { p n ( ⋅ ; a | q ) } n = 0 ∞ are classically defined for 0 q 1 and 0 a q 1 . After extending this family to a new one in which arbitrary real values of the parameter a are allowed, we give an orthogonality condition for those cases for which Favard’s Theorem fails to work.

Non-classical orthogonality relations for big and little q-Jacobi polynomials

Abstract Big q -Jacobi polynomials { P n ( ⋅ ; a , b , c ; q ) } n = 0 ∞ are classically defined for 0 a q − 1 , 0 b q − 1 and c 0 . For the family of little q -Jacobi polynomials { p n ( ⋅ ; a , b | q ) } n = 0 ∞ , classical considerations restrict the parameters imposing 0 a q − 1 and b q − 1 . In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditions for those cases for which Favard’s theorem does not work. As a by-product, we obtain similar results for the families of big and little q -Laguerre polynomials.

New Infinite Products of Cosines and Viète-Like Formulae

Summary In this article, the authors show that Viètes formula is only the tip of the iceberg. Under the surface, they search for curious and interesting Viète-like infinite products, rare species made of products of nested square roots of 2, but here with some minus signs occurring inside. To explore this fascinating world, they only use the simple trigonometric identity cos x = 2 cos((π + 2x)/4) cos ((π - 2x)/4), combined with a recent formula by L. D. Servi.

A complete view of Viète-like infinite products with Fibonacci and Lucas numbers

The main goal of this paper is to link the nth Fibonacci and Lucas numbers through certain infinite products of nested radicals. This work relies on recent results on Viete-like infinite products appeared in Moreno and Garcia-Caballero (2013) 3. We will analyze in detail one particular case of these formulas and we will show how our treatment covers and extends previous results in the literature.

Generating functions and companion symmetric linear functionals

AbstractIn this contribution we analyze the generating functions for polynomials orthogonal with respect to a symmetric linear functional u, i.e., a linear application in the linear space of polynomials with complex coefficients such that % MATHTYPE!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGae8xDau3aaeWaaeaacqWF4baEdaahaaWcbeqa% aGqbaiab+jdaYiab-5gaUjaabUcacaaIXaaaaaGccaGLOaGaayzkaa% Gaeyypa0JaaGimaaaa!44F8!

The unruly sin(π/7)

u\left( {x^{2n + 1} } \right) = 0

Chebyshev polynomials and nested square roots

. In some cases we can deduce explicitly the expression for the generating function % MATHTYPE!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFpepudaqadaqaamXv% P5wqonvsaeXbfv3ySLgzaGGbciab+Hha4jab+XcaSiab+Dha3bGaay% jkaiaawMcaaiabg2da9maaqahabaGae43yam2aaSbaaSqaaiab+5ga% UbqabaGccqGFqbaudaWgaaWcbaGae4NBa4gabeaakmaabmaabaGae4% hEaGhacaGLOaGaayzkaaGae43DaC3aaWbaaSqabeaacqGFUbGBaaGc% caGGSaaaleaacqGFUbGBiyaacqqF9aqpcaaIWaaabaqeduuDJXwAKb% Yu51MyVXgaiCaacqaFEisPa0GaeyyeIuoaaaa!64B4!