Representation of functions that are regular at the origin
In Chapter II, the function Ψ used in defining a set of generalized Appell polynomials was itself an entire function; the functions for which we obtained expansions were in the class ℝ Ψ , which is a class of entire functions. In this chapter we take up t
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Set
H(z, w)
47
00
=
L p,. (z) w"fn!
n=O
oo (z+ i)2nw2"
oo (z-i)2n+lw2n+l
=,.~0--(~+ ..~0
(2n+1)!
=
cosh(z + 1) w + sinh(z -1) w
=
ezw cosh w - e-zw sinh w
o
Although the polynomials are not generalized Appell polynomials, we can write cosh (2w) ezw = H(z, w) cosh w + H(z,- w) sinh w·
L p,.(z) {coshw + (-1)"sinhw}w"fn!. 00
=
n=O
The process that we used for expanding a function in a series of generalized Appell polynomials still applies, and we obtain the following theorem, which goes considerably beyond the results of NASSIF and EWEIDA.
Theorem 12.1. A ny entire function f of exponential type can be expanded in a convergent series of the form L .P" (!) Pn (z), where ::t'. (f) = ~1-. 11
2nt
J
r
w"{coshw+ (-1)"sinhw} n!cosh(2w)
F(w) dw,
F is the Borel transform of f, and F encloses the conjugate indicator diagram of f and does not pass through a zero of cosh (2 w). There are infinitely many different representations of zero, corresponding to the zeros of cosh(2w). The zeros nearest the originare at ±ni/4. Since the basic series is the expansion obtained when F encloses no zeros of cosh (2 w), the basic series is applicable to entire functions of exponential type less than n/4, and this critica} type cannot be replaced by any larger number. However, ali that is really needed is to have the conjugate indicator diagram avoid all the zeros, so that it is enough, for example, to have h(±n/2)O, we suppose that Iim 'P,.11"
n=O
exists (finite), and we suppose, by way of normalization, that thie limit is 1. Then 'P(t) is regular for ali t in a set E that contains the open disk 1ti< 1 and has t = 1 as a boundary point. In Chapter II ws singled out for study the class S'rop of functions f(z) = 1:/,.z11 of finite 'P-type, i.e. such that Iim sup j/,./'l',.l 1'" -1, {J> O. Since PJP·"'l(z) = (-1)" P~"'·Pl(z), this holds also if cx:> O, {J> -1. To obtain the complete known result 2 that this holds if cx: > -1, {J > - 1, we appeal to the fact that 2 :z p~at-l,P-ll(x)
=
(n
+ cx: + {J- 1) p~~fl(x),
so that by expanding f'(z) in terms of P~"'·Pl(z) we can deduce an expansion of l(z) in terms of PJ"'-l,P-ll(z). The restriction oc.> -1, {J> -1 is now seen to be irrelevant for the expansion of analytic functions; we are not using the orthogonality of the polynomials, which is all that is lost for smaller values of cx: and {J.
§ 18. Polynomials not in generalized Appell form Just as for expansions of entire functions, the general theory serves to elucidate the behavior of some sets of polynomials which are not in the form of generalized Appell polynomials. As an instance of this possibility, consider the set given by Po = 1, p,. (z) = 1 + zn (n >O); this is used as an example by WHITTAKER 3. The function
K(z, w) =
00
00
n=O
n=l
L p,.(z) wn = 1 + L {1 + zn) wn =
(1- zw)-1
+ w/(1- w)
has singularities at w = 1 and w = 1/z. If Iz 1 < r, thcn K is regular for 1 w 1 < 1/r except at w = 1. Thus the series expansion for K(z, w) converges for all lwl1, but only for lwlO, where the path of integration is u
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