ON A GENERALIZATION OF HANKEL KERNEL

We consider an expression involving the Bessel function, the Neumann function and the MacDonald function and discover various combinations of these functions which are Fourier kernels or conjugate Fourier kernels. Also a large number of integration formulae are established involving these kernels.


INTRODUCTION.
In a previous paper [1], we considered the manner in which Fourier Kernels may be generated as solutions of ordinary differential equations.We generated some previously known Fourier Kernels in this way and many others.One of the former ones was [2,9], 2 x))] k(x) [sin -J(x) + cos --(Y(x) + K,( involving the Bessel function J, the Neumann function Y,, and the MacDonald function K In this paper we follow a different line of thought.We inquire which expressions of the type k(x) [AJ(x) + BY,(x) + CK(x)] (for constant A, B, C) are Fourier kernels or have conjugate kernels of the same form.In this manner, we discover some new Fourier kernels and others which have a simple looking conjugate.
Also, we establish a large number of integration formulae, involving the function k(x) and its conjugate.Many of these formulae are believed to be unavailable in the literature.Throughout, we point out various known results as special cases of our general results.

PRELIMINARIES.
We shall mention below a few known results and definitions from the theory of Mellin transforms, which will be needed later.All of these results can be found in [3].A function F(s), c + it,-(R) < < (R), a < c < b, is said to be the Mellin transform of f(x), if I F(s) f(x) x s-1 dx If(x); s].
And conversely, we call, c/i.I F(s) x -s ds f(x) J/'t[F(s); x] the inverse Mellin transform of F(s).
An important result in the theory of Mellin transform, is the Parseval theorem: If F(s) and K(s) are the Mellin transforms of the functions f(x) and k(x) respectively, then, under appropriate conditions.c+ioo II I g- A direct consequence of the Parsev theorem is that if K(s)F(-s) a(s), (.) where K(s), F(s) and G(s) denote the Mellin transforms of k(x), f(x) and g(x) respectively, then, in a suitable strip of the s-plane, we have k(xt)f(t) dt g(x). (2.3) 2), and we cl g to be the k-transform of f.If further, the inversion formula involng the kernel h(x), holds, then k and h are said to be conjugate of each other.Also, their Mellin transforms satisfy the function equation K(s)( ) in some strip of the s-plane,.If instead of (2.4), we have the inversion forma [ k(xt)g(t) dt f(x) (.) ong with (2.3), then k is said to be selfonjugate or a Fourier kernel.Also its Mdlin trsform satisfies the equation K(s)K(1 s) 1. (2.6) Thus, if the equations (2.3) and (2.4) hold simtaneously, then.we shall cl k(x) and h(x), conjugate kernels.If, on the other hd, equations (2.3) and (2.5) hold, then k(x) is sd to be a sdfonjugate kernel.
Let k be a selfonjugate kernd.If for some stable f, then f is sd to dgenfunction of the operator k, corresponng to the eigenvMue 1 respectivdy.It shoed noted that if the operator k is a Fourier-kernel, then it has oy these two eigenvues.
3. THE KERNELS.We consider the function k(x) v [A Jr(x) + BYv(X) + CKv(X)], where A, B and C are real constants.One can assign appropriate values to these constants so that k(x) is either self-conjugate or has a conjugate of the same type.Our first task will be to determine those values of A, B and C. The technique we shall employ to find those suitable values, consists of using results from Mellin transform theory.
The crucial part of our procedure is to express the function K(s), the Mellin transform of k(x), as a rational expression of Gamma functions.
Now, making use of the Mellin transform of the functions ./fJu(x),,/fYu(x) and ;JKu(x), [4], the Mellin transform of k(x) is then given by K(s) ([k(x); s] where I1 < Re < 1.In order to consolidate the bracketed terms into a single term, an appropriate choice for the constants is that 2 A cos 8r, B sin 8r, C F sin where # and a are arbitrary.Then one can write after some simplification, K(s)=r 2s-F( + v + s)F(-v + s)sin (-v-20 + 2a + s).sin ( + v + 28 + 2a-s).Now using the functionM equation r(z) r(1 z) cosec and the duplication formula for r(2z), we obtMn, K(s) 2s-F( ..i (.)To determine the function h(x), the conjugate of k(x), we consider the function equation H(s)g(1 -s) 1 where H(s) and K(s) are the Mdn trsforms of h(x) and k(x) resctively.Whence, a(s) :- Now in a stable strip of the s-ple, we have h() -[H(s); ], wch can be shown, by complex integration, to the sum of two hypergmetric series, eventuly giving us the conjugate of the function k(x).
In the next two sections, we shall explore situations giving rise to four special cases.
These cases are of particular interest since they lead to a simpler representation of the conjugate function h(x).
In some instances h(x) coincides with k(x), defining a self-conjugate kernel.We shall discuss self-conjugate kernels first. 4. SELF-CONJUGATE KERNELS.
Again letting v 1 and 2, we obtain special cases of (4.9), which are respectively  where (s) (1 -s).
cases when In general, in order that f(x) should be an eigenfunction of the operator k corresponding to the eigenvalue 1, F(s), s a + i-, the Mellin transform of f should be of the form The eigenfunctions mentioned above in (4.4) and (4.9) are special and when W(s) 1 respectively.Then from (4.1), we have the functional equation K (s)F(1 s) G(s).
(5.27) a +t We wish to note that, to our knowledge, all the results for which we have not given references from the literature, appear to be new.Our method, therefore, has yielded a large number of new integration formulae.

SOME APPLICATIONS
Since the kernels in this paper are also solutions of a Fourth Order ordinary differential equation [1], it is expected that our results will find applications in situations which involve such differential equations.One such situation was encountered in [1].We point out some more below.
If we consider the problem of finding solutions of O4----u-I-ou 0 in 0 < x < (R), > 0 Ox4 or of +-+ =0 in 0<r<(R) t>0 r which solutions are bounded at infinity, and which satisfy the conditions 0u 0u 0) at x=0 (or at r=0) (1) u j 0 (or u -= or the conditions (6.1) ( (2) 02u 03u 0x Ox o (or Vu V2u=0) at x=0(or at r=0) 0r (6.4)   respectively, where V 0 1 0 Or r Or then we encounter the kernels introduced in this paper.If these solutions are subject to the initial conditions u g,() (6.5a) and -= g2() (6.5b) at t=0 in 0 < < (R), where is either x or r depending upon whether we are dealing with equation (6.1) or equation (6.2), then the solution is u(,t) f0 (R) k(A)[A() cos(,2t) +sin(2t)]d (6.6)where gt() f0 A(,)k(A)dA (6.7) and g2() J0 2B()k()d,k (6.8) where k is an appropriate kernel.If k is self-conjugate then the solution of equations (6.7) and (6.8)is A(A)--fO g* ()k(A)d and substitution in equation (6.6) gives u.The following cases should be noted: ou (1) If u= and the conditions are u j 0 at the origin, then equation (6.6)   gives the deflection of a vibrating semi-infinite elastic rod which is clamped at one end (the origin) and is subject to the initial conditions (6.5).In this case k kl(x (sin x cos x + e "x) which is self--conjugate.
(2) If u 1 / 2 and the conditions are 0u 03u Ox 3 0 at the origin, then equation (6.6) gives the deflection of a vibrating semi-infinite elastic rod which is free at one end (the origin) and is subject to the initial conditions (6.5).In this case k ks(x (cos xsin x + e -x) which is self-conjugate.

Ou
(3) If u=--0 and the conditions are u -0 at r=0, then equation (6.6) gives the deflection of a (symmetrically) vibrating infinite elastic plate which is damped at the origin and is subject to the initial conditions (6.5).
(6.13) "0 Since k is self conjugate, these equations are easily inverted and then substitution gives u.