Measures of Growth of Entire Solutions of Generalized Axially Symmetric Helmholtz Equation

For an entire solution of the generalized axially symmetric Helmholtz equation , measures of growth such as lower order and lower type are obtained in terms of the Bessel-Gegenbauer coefficients. Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients.


Introduction
e solutions of the partial differential equation are called the generalized axially symmetric Helmholtz equation functions (GASHEs). e GASHE function , regular about the origin, has the following Bessel-Gegenbauer series expansion: where = cos , = s , + are Bessel functions of �rst kind, and are Gegenbauer polynomials. A GASHE function is said to be entire if the series (2) converges absolutely and uniformly on the compact subsets of the whole complex plane. For being entire, it is known [1, page 214] that: �ow we de�ne Following the usual de�nitions of order and type of an entire function of a complex variable , the order and type of are de�ned as ( ) = l m sup ∞ log log ( , ) log , Gilbert and Howard [2] have studied the order ( ) of an entire GASHE function in terms of the coefficients 's occurring in the series expansion (2) (see also [1, eorem 4.5.9]). It has been noticed that the coefficients characterizations for lower order and lower type of have not been studied so far. In this paper, we have made an attempt to bridge this gap. McCoy [3] studied the order and type of an entire function solutions of certain elliptic partial differential equation in terms of series expansion coefficients and approximation errors. Recently, Kumar [4,5] obtained some bounds on growth parameters of entire function solutions of Helmholtz equation in 2 in terms of Chebyshev polynomial approximation errors in sup-norm. In the present paper, we have considered the different partial differential equation from those of McCoy [3] and Kumar [4,5] and obtained the growth parameters such as lower order and lower type of entire GASHE function in terms of the coefficients in its Bessel-Gegenbauer series expansion 2 Journal of Complex Analysis (2). Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients. Our approach and method are different from all these of the above authors.

Auxiliary Results
In this section, we shall prove some preliminary results which will be used in the sequel.
We prove the following lemma.

Lemma 1.
If is an entire GASHE function, then for all > 0, * > 1, * < ∞ and for all , Proof. First we prove right hand inequality. Using the relations in (2), we get Now to prove le hand inequality, we use the orthogonality property of Gegenbauer polynomials [1, page 173] and the uniform convergence of the series (2) as From the series expansion of + ( ), we get and for where [ ] denotes the integral part of , we have From (7), (9), and (11), for Since (12) gives that Hence the proof of Lemma 1 is complete.
We now de�ne Lemma 2. If is an entire GASHE function, then and are also entire functions of the complex variable . Further Proof. Let be an entire. In view of (3), we have Hence both and are entire. Inequalities in (15) follow from (6). (14). en orders and types of and , respectively, are identical.
Similarly, using Lemma 5 and (48) for entire function e result (46) is now followed by (24) and above two relations for ( ) and ( ).
If ( ) = ∑ =0 is an entire function of order ( ), lower type ( ), and | / +1 | forms a nondecreasing function of for > 0 , then by a result of Shah [11], we have Equation (47) now follows in view of (22) and (25). Hence the proof is complete.