On a theorem of Pólya on entire functions with real Taylor coefficients

1997 ◽  
Vol 38 (1) ◽  
pp. 37-46 ◽  
Author(s):  
A. M. Gaîsin
Keyword(s):  
Entire Functions ◽  
Taylor Coefficients

scholarly journals On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
M. Abul-Ez ◽  
H. Abd-Elmageed ◽  
M. Hidan ◽  
M. Abdalla
Keyword(s):  
Entire Functions ◽  
Maximum Modulus ◽  
Growth Order ◽  
Complex Matrix ◽  
Growth Type ◽  
Complex Matrices ◽  
Taylor Coefficients ◽  
Linear Substitution ◽  
Higher Dimensional ◽  
Explicit Relation

In this paper, we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of entire functions in several complex matrix variables (FSCMVs) in hyperspherical regions. The obtained formulas enable us to compute the growth order and the growth type of some higher dimensional generalizations of the exponential, trigonometric, and some special FSCMVs which are analytic in some extended hyperspherical domains. Furthermore, a result concerning linear substitution of the mode of increase of FSCMVs is given.

scholarly journals On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients

2021 ◽  
Vol 56 (2) ◽  
pp. 149-161
Author(s):  
T. H. Nguyen ◽  
A. Vishnyakova
Keyword(s):  
Entire Function ◽  
Theta Function ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Taylor Coefficients ◽  
Partial Theta Function ◽  
Necessary And Sufficient

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.

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