A treatise on the theory of Bessel functions, by G. N. Watson.

CHAPTER IX

POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONS
9 1. The definition of N7eumann's polynomial 0,, (t).
The object of this chapter is the discussion of certain polynomials which
occur in various types of investigations connected with Bessel functions.
The first of these polynomials to appear in analysis occurs in Neumann's *
investigation of the problem of expanding an arbitrary analytic function f(z)
into a series of the form Ya,,J, (z). The function 0,, (t), which is now usually
called Neumann's polynomial, is defined as the coefficient of e,,J (z) in the
expansion of 1/(t - z) as a series of Bessel coefficients-t, so that
1
(1) t,- aJO (z) 0 (t) + 2J, (z) 0 (t) + 2J, (z) 0 (t) +
c -
e 2 J1 (z OQ) (t).
1==0
From this definition we shall derive an explicit expression for the function,
and it will then appear that the expansion (1) is valid whenever z < t.
In order to obtain this expression, assume that z < i t and, after expanding
1/(t - z) in ascending powers of z, substitute Schlomilch's series of Bessel
coefficients (~ 2'7) for each power of z.
This procedure gives
1 1 ' zs
-+ +
t-z t S,- t+1
E 1 2Z>S () +%~ + T E2s (s 2n). (s + +. -1)!
=, e2 6. (Z7)+ ' C 2 -— J +2n (Z),
t =o s=l =0 i
Assuming for the moment that the repeated series is absolutely convergent j,
' Theorie der Bessel'schen Functionen (Leipzig, 1867), pp. 8-15, 33; see also Journal fiOr
Mlath. LXVII. (1867), pp. 310-314. Neumann's procedure, after assuming the expansion (1), is to
derive the differential equation which will be given subsequently (~ 9'12) and to solve it in
series.
+ In anticipation of ~ 16'11, we observe that the expansion of an arbitrary function is obtained
by substituting for 1/(t - z) in the formula
f () 1 f+ )f (t)dt
+ Cf. Pincherle's rather more general investigation, Rendiconti R. Ist. Lomlbardo, (2) xv. (1882),
pp. 224-225.