IDENTITIES INVOLVING ITERATED INTEGRAL TRANSFORMS

A number of identities involving iterated integral transforms are established, 
making use of the fact that a function which is a linear combination of the Macdonald's function Kν(z), where z is a complex variable, is a Fourier kernel.


i. INTRODUCTION.
The object of this note is to establish various identities involving integral operators.The integral operators are the integral transforms with respect to the function K (z), where 7/ (z) is the Macdonald's function of order and argument z, a complex variable.Some functional relations are deduced, as special cases, which show the inter-relations among more familiar Fourier Sine, Fourier Cosine, and Laplace transforms.

THE KERNEL.
Let 1/27/ (0), with a constant and vl < I. x--'---y (-I) y (2.1)It is not difficult to see that if x is a complex variable, then every point of (2.1) is regular except for a singularity at x 0. Now consider a function of the form 2k-1 The functions are an extension of the functions which were first noted by Guinand.
As a special case when k 2, chose the coefficients as Then we obtain and we have THEOREM 2.1.y kv(x) is a solution of .Z114 0 < X <oo the two-fold Bessel equation.
The function k (x) is of special interest to us here nd we shall develop its properties further.
These functions arise as kernels in divisor summation formulae of the Hardy-Landau type, involving number theoretic function ok(n), the number of kth powers of the 1 divisor of n, [4] If we put v have -2 ,we k+1/2(x) -(cos x -sin x + e which obviously satisfies the differential equation Dy y. I. Next, the Mellin transform of xK (x) is given by where Re s > IRe w -- [5] whence the Mellin transform of k (x) defined in (2 2) is given by On simplifying, we have 2s+ k*(s)

L2
Now, if we take (x) (0,), the integral defining the transformation T exists and is in fact absolute convergent.Thus T is a bounded transformation on L2-space for Ivl < i.
We shall now define the transformation T in operator notation.Denote the operators K and K respectively by where f L 2 (0,) and I,I < , with K (z) being the Macdonald' s function.Then the transformation can be expressed, in operator form, as and we can write, symbolically, T---(K.,i + K,_z.+ 2cos vK Since F 2 f, the identity transformation, we have 1 v,-.
" K2 + K .K + K .K + 2cos-v(g .K + K .K + K K + K K The right-hand side is the linear combination of iterated transformations, which are (3.1)bounded on L2-space for vj I. Now, using the standard result [5], I tKv(at)Kv(Bt)dt where Ivl < 1 and Re(u+B) > 0, we have for example, if 2(0,), The change of order of integration can be justified by absolute convergence.Thus, we obtain our first identity The identity given by (3.2) can alternatively be established by making use of the Mellin transform theory.That is, the Mellin transform of the iterated operator Kv,iKv[f], is given formally by m{K where f*(s) denotes the Mellin transform of f(x).Also, [f;} m{-1 1/2 x K (x); s}m{-1 x1/2K (-ix); l-s}f*(s) K .K +KK =o as shown above.Similarly, one can show that KK .+K.K =o, a sort of conjugate of the identity in (3.2).Consider the representation ( (e-i1/2vJ (x) e i1/2w (x)); By comparing the real and imaginary parts and solving, we obtain the identities KH =H K Now, going back to equation (3.1) and using the results given by (3.2), (3.3), and (3.9) in (3.11) and simplifying, we have finally, for Ivl i, ) --) --) ) (3.12) An interesting relationship can be established by putting v i in (3.12).Then where ,C, and L denote the Fourier sine, Fourier consine and Laplace transforms respect ively.
From (3.9) and (3.12), one can establish the identity K 2 _K _K 2 ,i w,-i which, on setting v -+ i , yields (3.13). (3.14) IRe I --By repeated use of the relation r(z)r(1-z) sin zz it is not a difficult matter to see that k*(s)k(l-s) I. Hence, THEOREM 22.The function k*(x) defined by equation (2.2) is a Fourier kernel, [6] If we define the transformation T[f] by c[f] I] k(xt)f(t)dt, then C2[f] f and T is involutory since F2 7, the identity transformation.Mak- ing use of the asymptotic expansions of the Macdonald's function K (z), we have and k 2vH If] ei1/2H []}where H denotes the Hankel transform operator of order v. Thus, in operator form, )Hw) e (HK KH_v).
f] C[f],where S and C are the usual Fourier since and cosine transform repsectively.Also,K1/2[f] I (xt)1/2K1/2(xt)f(t)dt I11/2 Ie-Xtf(t)dt c[yl,where L[f] is the Laplace transform.general relation involving the operators K and H can be established: i HK cosec { sin-(w-) KvH + sin -g = ( 1 T[identity operator, with being the Hankel transform.Similarly, one can show that (3.10)And, in the same vein, we have