Ladder-operator Factorization and the Bessel Differential Equations

We present an alternative approach to the discussion of Bessel equations and Bessel functions, through an elementary factorization method. The various Bessel equations are represented by a single parameterized form and, after a standard transformation of the dependent variable, a transformed parameterized (Bessel) equation is factorized in terms of raising and lowering ladder-operators. Once constructed, the ladder-operators for the transformed parameterized equation determine the ladder-operators that factorize the various Bessel equations and enable the determination of the various recurrence relations between the Bessel functions. In particular the construction of the Rayleigh formulae for the Bessel functions becomes particularly straightforward. However, 'starting' Bessel functions for the ladder operators and iterative and Rayleigh formulae must still be obtained as series solutions of particular Bessel equations.


Introduction
Arising from many important problems in science and engineering, the Bessel equations form one of the linchpins of the theory of special functions [4].It is W. Robin usual for the various Bessel functions, and their properties, arising from the solution of the Bessel equations to be developed either from the well-known series solutions of the Bessel equations [6], or their complex integral representation [4].
Here, however, we offer an alternative approach to the solution of the various Bessel equations, and the properties of the resultant Bessel functions, through an application of a factorization technique.
To describe the basic idea, we represent the generic Bessel function by ) (x Z n and consider the generic Bessel equation to be of the form makes our starting point marginally more general. which form suggest that we assume that (1.3) can be written in factorized forms, in terms of raising and lowering operators [3].. This programme is developed below, in subsequent sections.
Finally, before we consider the details of the method, we note that the Riccati-Bessel equation 0 [1] and will not be considered further here.

The General Method
Our discussion above suggests looking for raising and lowering operators (with the s ' when we assume that (1.3) can be written in the factorized forms [3] ) By comparing (1.5) with (2.2), we hope to determine the unknown functions ), (x p ), (x q , + n ϕ and .
− n ϕ We find that the following sets of consistency conditions must be satisfied: and 0

W. Robin
It is apparent that we may eliminate between (2.3a) and (2.3c) and between (2.3a) and (2.3d) to get two Riccati equations, one for ) (x p and one for ), (x q that is 0 It is a straightforward matter to pick-out particular solutions to equations (2.4), especially when we keep (1.5) and (2.3a) and (2.3b) in mind.The general solutions, should they be required, can then be written down straight away [6].Indeed, by inspection, we find equations (2.4a,b) satisfied identically if We now have the transformed Bessel equation, (1.3), in factorized form(s), (2.2), and have developed a ladder-operator formalism, (2.1), for the corresponding transformed Bessel functions, )

. (x w n
To complete the factorization or ladder-operator formalism for the original Bessel equation, (1.1), and its corresponding Bessel function, ), (x Z n we (a) substitute back using (1.2) and obtain the formalism in terms of the original set-up, and (b) determine a particular solution ) (x Z n for any integer n (usually .0 = n ) The ladder-operator formalism enables us to find all other ) (x Z n from the given particular function (which particular function will be a solution in series).Further, the ladder-operator formalism enables the development of other relations between the Bessel functions, like the three-term recurrence relations and the Rayleigh formulae, and so on.Note that each individual Bessel equation has two fundamental particular solutions associated with it for each value of , n so each ladder-operator representation has two starting points: one starting point for each set of fundamental particular solutions.In addition, other functions defined in terms of the two fundamental particular solutions, the Hankel functions say [4],will also satisfy the same fundamental relations as the defining Bessel functions.
The series solution to (1.3) is easy to obtain.However, all that we require is (2.6c)From (2.6b) and (2.6c), we see that all odd-numbered coefficients vanish and we are left with even-numbered coefficients only.If we shift the dummy index r down two units and replace r with , 2m then the non-zero coefficients satisfy the adjusted recurrence relation On the other hand, when The second solutions corresponding to (2.9b) and (2.10b), along with other relevant functions, are obtained in a standard manner as discussed, briefly in section 5 below.

The Basic Bessel Equations
Suppose the Bessel function ) (z Z n is a particular solution of a Bessel equation of order , n or satisfy the differential equation (see, also, Bernardini and Natalini [2]) To find the ladder-operators for (3.3), we substitute 1 = k into equations (2.4) and solve the resulting two Riccati equations to get, in particular, from (2.5) in agreement with the results of Bernardini and Natalini [2].
The differential recurrence relations (3.4), on substituting 2), reduce to their 'usual' forms [6] Ladder-operator factorization 77 where we have replaced n with 1 − n in the final form of equation (3.5b).We may now factorize the Bessel's equation of (integral) order n, since, from equations (3.5), it follows that equation (3.1) is identical to (for example) We may infer further relations from equations (3.5).For example, on eliminating the derivative terms from equations (3.5), we find that which is the 'usual' [6] n an integer and x a real variable.The constants , a , b c and k determine the particular type of Bessel equation (and Bessel function) and are defined when each Bessel equation is encountered below, in sections 3 and 4.The general method is developed as follows.If ) (x Z n is one of the Bessel functions, then we consider a Lommel transformation of the dependent variable in the corresponding Bessel equations of the form[2

d
remaining as an arbitrary coefficient.Interestingly, k takes on the two values 1 and 2 only (see sections 3 Bessel equation.Following our discussion in sections 1 and 2, we compare (3.1) with (1.1) and note that for(3

3 )
equation.Following our discussion in sections 1 and 2, we compare (3.1) with (1.1) and note that for (3To find the ladder-operators for (4.3), we substitute 2 = k into equations (2.4) and solve the resulting two Riccati equations to get, in particular, from(2 three term recurrence relation for Bessel functions.Or, again, if we examine (3.5a) we have, by mathematical induction