These two pages give the short notes about the properties and recurrence relations

1. Mathematical Physics
1.1 Bessel Function
Bessel differential equation is given as
𝑥2
𝐽𝑣
′′( 𝑥) + 𝑥𝐽𝑣
′ ( 𝑥) + ( 𝑥2
− 𝑣2) 𝐽𝑣 = 0
The generating function
𝑔( 𝑥, 𝑡) = 𝑒𝑥𝑝 [
𝑥
2
( 𝑡 −
1
𝑡
)] = ∑ 𝐽𝑛 (𝑥)𝑡 𝑛
∞
𝑛=−∞
The corresponding solution (Bessel function – First kind)
𝐽𝑛
( 𝑥) = ∑
(−1) 𝑠
𝑠! ( 𝑛 + 𝑠)!
(
𝑥
2
)
𝑛+2𝑠
∞
𝑠=0
Properties:
 𝐽−𝑛
( 𝑥) = (−1) 𝑛
𝐽𝑛(𝑥)
 𝐽0
′ ( 𝑥) = −𝐽1(𝑥)

𝑑
𝑑𝑥
[ 𝑥 𝑛
𝐽𝑛(𝑥)] = 𝑥 𝑛
𝐽𝑛−1(𝑥)

𝑑
𝑑𝑥
[ 𝑥−𝑛
𝐽𝑛(𝑥)] = −𝑥−𝑛
𝐽𝑛−1(𝑥)

𝑑
𝑑𝑥
𝐽0
( 𝑥) = −𝐽1(𝑥)
 𝐽𝑛
( 𝑥) = 𝐽𝑛+1
′ ( 𝑥) +
𝑛+1
𝑥
𝐽𝑛+1(𝑥)
 𝐽𝑛
( 𝑥) = −𝐽𝑛−1
′ ( 𝑥) +
𝑛−1
𝑥
𝐽𝑛−1(𝑥)
 𝐽0
( 𝑥) =
1
𝜋
∫ cos( 𝑥 sin 𝜃)
𝜋
0
𝑑𝜃
 cos 𝑥 = 𝐽0
( 𝑥) +
2 ∑ (−1) 𝑛
𝐽2𝑛
( 𝑥)∞
1
 sin 𝑥 = 2 ∑ (−1) 𝑛
𝐽2𝑛+1
( 𝑥)∞
1
 𝐽1
2
( 𝑥) = √
2
𝜋𝑥
sin 𝑥
 𝐽−
1
2
( 𝑥) = √
2
𝜋𝑥
cos 𝑥
 𝐽−1
( 𝑥) + 𝐽1
( 𝑥) = 0

2. 2
Recurrence relation: 𝐽𝑛−1
( 𝑥) + 𝐽𝑛+1
( 𝑥) =
2𝑛
𝑥
𝐽𝑛(𝑥)
𝐽𝑛−1
( 𝑥) − 𝐽𝑛+1
( 𝑥) = 2𝐽𝑛
′
(𝑥)
𝑥𝐽𝑛
′
(𝑥) = 𝑛𝐽𝑛(𝑥) − 𝑥𝐽𝑛+1(𝑥)
2𝐽𝑛
′ ( 𝑥) = 𝐽𝑛−1
( 𝑥) − 𝐽𝑛+1(𝑥)
Orthogonality:
∫ 𝑥 𝐽𝑛
( 𝛼𝑥)
1
0
𝐽𝑛
( 𝛽𝑥) 𝑑𝑥 = 0
Bessel function – Second kind: Neumann function:
𝑌𝑣
( 𝑥) =
cos 𝑣𝜋 𝐽𝑣
( 𝑥) − 𝐽−𝑣(𝑥)
sin 𝑣𝜋
𝑌𝑛−1
( 𝑥) + 𝑌𝑛+1
( 𝑥) =
2𝑛
𝑥
𝑌𝑛(𝑥)
𝑌𝑛−1
( 𝑥) − 𝑌𝑛+1
( 𝑥) = 2𝑌𝑛
′
(𝑥)
Bessel function – Third kind: Hankel function:
𝐻𝑣
1( 𝑥) = 𝐽𝑣
( 𝑥) + 𝑖𝑌𝑣 (𝑥) and 𝐻𝑣
2( 𝑥) = 𝐽𝑣
( 𝑥) − 𝑖𝑌𝑣 (𝑥)
𝐻𝑛−1
( 𝑥) + 𝐻𝑛+1
( 𝑥) =
2𝑛
𝑥
𝐻𝑛(𝑥)
𝐻𝑛−1
( 𝑥) − 𝐻𝑛+1
( 𝑥) = 2𝐻 𝑛
′
(𝑥)