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We study extension of $p$ -trigonometric functions ${sin}_{p}$ and ${cos}_{p}$ to complex domain. For $p = 4, 6, 8, \dots$ , the function ${sin}_{p}$ satisfies the initial value problem which is equivalent to (*) $- {(u^{'})}^{p - 2} u^{''} - u^{p - 1} = 0, u (0) = 0, u^{'} (0) = 1$ in $ℝ$ . In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_{p} (x)$ is a real analytic function for $p = 4, 6, 8, \dots$ on $(- π_{p} / 2, π_{p} / 2)$ , where $π_{p} / 2 = \int_{0}^{1} {(1 - s^{p})}^{- 1 / p}$ . This allows us to extend ${sin}_{p}$ to complex domain by its Maclaurin series convergent on the disc ${z \in ℂ : | z | < π_{p} / 2}$ . The question is whether this extensions ${sin}_{p} (z)$ satisfies (*) in the sense of differential equations in complex domain. This interesting...

Grundlagen einer independenten Theorie der elliptischen Modulfunctionen und Theorie der Multiplicatorgleichungen erster Stufe.

Hurwitz (1881)

Mathematische Annalen

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Formulae novae in theoria transcendentium ellipticarum fundamentales.

Fourier-Bessel functions of singular continuous measures and their many asymptotics.

From holonomy of the Ising model form factors to $n$ -fold integrals and the theory of elliptic curves.

Functional Analysis, Orthogonal Polynomials And A Theorem Of Markov.

Functional equation of a special Dirichlet series.

Functional identities for special functions of quasiconformal theory.

Further results on supernumary polylogarithmic ladders.

General solution of overdamped Josephson junction equation in the case of phase-lock.

Généralisation des fonctions doublement périodiques de seconde espèce

Généralisation des fonctions $X_{n}$ de Legendre au cas de deux entiers, ou des fonctions qui naissent du développement des expressions ${[1 - 2 α x + α^{2}]}^{\pm \frac{2 l + 1}{2}}$ .

Generalisation of Bell polynomials and related operational formulas.

Generalization of two identities in Ramanujan's lost notebook

Generalizations of Bernoulli numbers and polynomials.

Generalizations of Euler numbers and polynomials.

Generalized Angelescu polynomial

Generalized trigonometric functions in complex domain

Grundlagen einer independenten Theorie der elliptischen Modulfunctionen und Theorie der Multiplicatorgleichungen erster Stufe.

Higher genus abelian functions associated with cyclic trigonal curves.

Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type III: The Heun case.

Holomorphic solutions of an inhomogeneous Cauchy equation.

Currently displaying 101 – 120 of 373