Hypertranscendency of perturbations of hypertranscendental functions

Abstract

Inspired by the work of Bank on the hypertranscendence of Γe h where Γ is the Euler gamma function and h is an entire function, we investigate when a meromorphic function f e g cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where f and g are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Bank's Conjecture (1977) on the hypertranscendence of Γe h. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form f + g, f • g and f • g in Theorem 3 and 4.

Linear 𝑞-difference, difference and differential operators preserving some 𝒜-entire functions

“…Finally, by M k we mean the field of meromorphic functions f with T (r, f ) = o(r k ) as r → ∞ outside a set of finite measure and by C[z] we mean the ring of polynomials with complex number coefficients. The field M k appears naturally in the studies of some hypertranscendental functions (see for example [17]). Using some ideas from Eremenko and Rubel [14] and Ng and Yang [26], and the half-plane version of Borel's lemma by Rossi [32] (see Lemma 2.2), we obtain the following Theorem 1.2.…”