Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

286

bearing in mind that Bessel polynomials J n ( z ) are generated by the function

∞ ∑ n = − ∞

G ( z , t )= e z

1 t ) =

2 ( t −

n .

J n ( z ) t

(5.267)

Solution Differentiating the equation (5.267) by z yields

1 t ! =

1 2

1 t

z 2 t −

∞ ∑ n = − ∞

n ,

J ′ n ( z ) t

t −

e

that is

1 2

1 t

∞ ∑ n = − ∞

∞ ∑ n = − ∞

n =

n

J ′ n ( z ) t

t −

J n ( z ) t

or 1 2

n − 1 ! =

∞ ∑ n = − ∞

∞ ∑ n = − ∞

∞ ∑ n = − ∞

n + 1 −

n .

J ′ n ( z ) t

J n ( z ) t

J n ( z ) t

All sums are reduced to t n , and given that n takes values from − ∞ to ∞ , we obtain 1 2 ∞ ∑ n = − ∞ J n − 1 ( z ) t n − ∞ ∑ n = − ∞ J n + 1 ( z ) t n ! = ∞ ∑ n = − ∞ J ′ n ( z ) t n . Equating coefficients next to t n we obtain

1 2

z ) − J n + 1 ( z ))= J ′ n ( z ) .

( J n

− 1 (

Problem 201

Prove the following identities

n z n z

J n ( z )+ J ′ n ( z ) ,

J n

z )=

− 1 (

J n + 1 ( z )= J n ( z ) − J ′ n ( z ) , bearing in mind that the Bessel function can be represented by the series

∞ ∑ r = 0

( z / 2 ) n + 2 r r ! ( n + r ) !

( − 1 ) r

J n ( z )=

(5.268)

.