Preliminaries
This chapter includes definitions and properties of Jackson q-difference and q-integral operators, q-gamma and q-beta functions and finally q-analogues of Laplace and Mellin integral transforms.
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Preliminaries
Abstract This chapter includes definitions and properties of Jackson q-difference and q-integral operators, q-gamma and q-beta functions and finally q-analogues of Laplace and Mellin integral transforms.
1.1 Some Classical Results In this section, we collect results from complex analysis which we shall use in this book. We will also introduce the definitions and terminology used in the text. Let f; g be entire functions and a 2 C, we say that f .z/ D O g.z/ ; as z ! a; ı if f .z/ g.z/ is bounded in a neighborhood of a. We write f .z/ g.z/; as z ! a; if lim
z!a
f .z/ D 1: g.z/
P n If f .z/ WD 1 nD0 an z is an entire function, then the maximum modulus is defined for r > 0 by M.rI f / WD sup fjf .z/j W jzj D rg : (1.1) The order of f , .f /, is, cf. [57, 128, 181], .f / WD lim sup r!1
log log M.r; f / n log n D lim sup : log r log jaj1 n!1 n
(1.2)
Theorem 1.1. [57, 122]. If f is entire and .f / is finite and is not equal to a positive integer, then f has infinitely many zeros, or it is a polynomial. The following two results of P´olya, cf. [237,238], concerning the zeros of cosine and sine transforms, are essential in our investigations. M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, DOI 10.1007/978-3-642-30898-7 1, © Springer-Verlag Berlin Heidelberg 2012
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1 Preliminaries
Theorem 1.2. Let f .t/ be a real-valued twice continuously differentiable function on the interval Œ0; 1. If jf .1/j > jf .0/j, the entire functions Z
Z
1
U.z/ D
f .t/ cos.zt/ dt;
V .z/ D
0
1
f .t/ sin.zt/ dt 0
have infinitely many real zeros and a finite number of complex zeros. Theorem 1.3. [238]. If the function f .t/ 2 L1 .0; 1/ is positive and increasing, then the zeros of the entire functions of exponential type Z 1 Z 1 U.z/ D f .t/ cos.zt/ dt; V .z/ D f .t/ sin.zt/ dt 0
0
are real, infinite and simple. The even function U.z/ has no zeros in Œ0; 2 /, and its positive zeros are situated in the intervals .k =2; k C =2/, 1 6 k < 1, one zero in each interval. The odd function V .z/ has only one zero z D 0 in Œ0; /, and its positive zeros are situated in the intervals .k; .k C 1//, 1 6 k < 1, one zero in each interval. The following version of Hurwitz–Biehler theorem for entire functions of order zero, cf. [181, Chap. 7] will be used in the sequel. Theorem 1.4. Let F .z/ be an entire function of order zero and assume that F .z/ D P .z/ C iQ.z/; where P .z/ and Q.z/ are entire functions with real coefficients. All roots of F .z/ lie in the upper half plane, =z > 0, if and only if P .z/ and Q.z/ have real, simple and interlacing zeros. Katkova and Vishnyakova[166] derived the following interesting result. P k Theorem 1.5. Assume that F .z/ WD 1 kD0 ak z is an entire function, ak > 0 for all k 2 N0 , and x0 is the unique positive root of the polynomial x 3 x 2 2x 1, x0 2:1479. Then x0 is smallest possible constant such that if ak akC1 > x0 ak1 akC2 .k 2 N/
(1.3)
then the zeros of F .z/ have negative real parts. Consequently,
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