In mathematics, a Neumann polynomial, introduced by Carl Neumann for the special case α = 0 , is a polynomial in 1/z used to expand functions in term of Bessel functions.
The first few polynomials are
O 0 ( α ) ( t ) = 1 t , O 1 ( α ) ( t ) = 2 α + 1 t 2 , O 2 ( α ) ( t ) = 2 + α t + 4 ( 2 + α ) ( 1 + α ) t 3 , O 3 ( α ) ( t ) = 2 ( 1 + α ) ( 3 + α ) t 2 + 8 ( 1 + α ) ( 2 + α ) ( 3 + α ) t 4 , O 4 ( α ) ( t ) = ( 1 + α ) ( 4 + α ) 2 t + 4 ( 1 + α ) ( 2 + α ) ( 4 + α ) t 3 + 16 ( 1 + α ) ( 2 + α ) ( 3 + α ) ( 4 + α ) t 5 . A general form for the polynomial is
O n ( α ) ( t ) = α + n 2 α ∑ k = 0 ⌊ n / 2 ⌋ ( − 1 ) n − k ( n − k ) ! k ! ( − α n − k ) ( 2 t ) n + 1 − 2 k , they have the generating function
( z 2 ) α Γ ( α + 1 ) 1 t − z = ∑ n = 0 O n ( α ) ( t ) J α + n ( z ) , where J are Bessel functions.
To expand a function f in form
f ( z ) = ∑ n = 0 a n J α + n ( z ) for | z | < c compute
a n = 1 2 π i ∮ | z | = c ′ Γ ( α + 1 ) ( z 2 ) α f ( z ) O n ( α ) ( z ) d z , where c ′ < c and c is the distance of the nearest singularity of z − α f ( z ) from z = 0 .
An example is the extension
( 1 2 z ) s = Γ ( s ) ⋅ ∑ k = 0 ( − 1 ) k J s + 2 k ( z ) ( s + 2 k ) ( − s k ) or the more general Sonine formula
e i γ z = Γ ( s ) ⋅ ∑ k = 0 i k C k ( s ) ( γ ) ( s + k ) J s + k ( z ) ( z 2 ) s . where C k ( s ) is Gegenbauer's polynomial. Then,
( z 2 ) 2 k ( 2 k − 1 ) ! J s ( z ) = ∑ i = k ( − 1 ) i − k ( i + k − 1 2 k − 1 ) ( i + k + s − 1 2 k − 1 ) ( s + 2 i ) J s + 2 i ( z ) , ∑ n = 0 t n J s + n ( z ) = e t z 2 t s ∑ j = 0 ( − z 2 t ) j j ! γ ( j + s , t z 2 ) Γ ( j + s ) = ∫ 0 ∞ e − z x 2 2 t z x t J s ( z 1 − x 2 ) 1 − x 2 s d x , the confluent hypergeometric function
M ( a , s , z ) = Γ ( s ) ∑ k = 0 ∞ ( − 1 t ) k L k ( − a − k ) ( t ) J s + k − 1 ( 2 t z ) ( t z ) s − k − 1 and in particular
J s ( 2 z ) z s = 4 s Γ ( s + 1 2 ) π e 2 i z ∑ k = 0 L k ( − s − 1 / 2 − k ) ( i t 4 ) ( 4 i z ) k J 2 s + k ( 2 t z ) t z 2 s + k , the index shift formula
Γ ( ν − μ ) J ν ( z ) = Γ ( μ + 1 ) ∑ n = 0 Γ ( ν − μ + n ) n ! Γ ( ν + n + 1 ) ( z 2 ) ν − μ + n J μ + n ( z ) , the Taylor expansion (addition formula)
J s ( z 2 − 2 u z ) ( z 2 − 2 u z ) ± s = ∑ k = 0 ( ± u ) k k ! J s ± k ( z ) z ± s (cf.) and the expansion of the integral of the Bessel function
∫ J s ( z ) d z = 2 ∑ k = 0 J s + 2 k + 1 ( z ) are of the same type.
