32.7. OTHER PROPERTIES OF BESSEL FUNCTIONS 631

We multiply these two series. This will involve many terms which can be added in anyorder thanks to absolute convergence. To get tm for m ≥ 0, you need to multiply termsl = m+ k times the term for t−k in the second sum. Thus you get for this term

tm∞

∑k=0

1(m+ k)!

( x2

)m+k 1k!

(−1)k( x

2

)k

= tm∞

∑k=0

(−1)k 1k!(m+ k)!

( x2

)2k+m= tmJm (x)

This gives the terms tm for m≥ 0.What of the terms involving m < 0? To get these terms, you need to have l− k = m so

you need k = l−m. Thus the sum which results for these terms is

tm∞

∑l=0

1l!

( x2

)l 1(l−m)!

(−1)l−m( x

2

)l−m= tm

∞

∑l=0

(−1)l−m

l!(l−m)!

( x2

)2l−m

= (−1)m tm∞

∑l=0

(−1)l

l!(l−m)!

( x2

)2l−m= (−1)m tmJ−m (x)

Therefore,e

xt2 e−

x2t = e(x/2)(t−1/t)

must equal the sum of tm terms for m≥ 0 and the sum of tm terms for m < 0. It follows that

e(x/2)(t−1/t) = J0 (x)+∞

∑m=1

tmJm (x)+∞

∑m=1

(−1)m t−mJm (x)

= J0 (x)+∞

∑m=1

Jm (x)(tm +(−1)m t−m)

Now recall that J−m (x) = (−1)m Jm (x) and so

e(x/2)(t−1/t) = J0 (x)+∞

∑m=1

Jm (x) tm +∞

∑m=1

t−mJ−m (x) =∞

∑m=−∞

tmJm (x)

That is, Jm (x) is just the mth coefficient of the series for e(x/2)(t−1/t). This has provedthe following interesting result on the generating function for Bessel equations.

Theorem 32.7.1 For m an integer and Jm (x) = (−1)m J−m (x) , we have the following gen-erating function for these Bessel functions.

e(x/2)(t−1/t) =∞

∑m=−∞

tmJm (x) (32.23)

In addition to this, there is an addition formula

Jm (x+ y) =∞

∑k=−∞

Jm−k (x)Jk (y) (32.24)