This document provides summaries of common special functions and polynomials, including Legendre polynomials, associated Legendre functions, Bessel functions, spherical Bessel functions, Hermite polynomials, and Laguerre polynomials. It defines the differential equations that govern each function, provides generating functions, discusses orthogonality properties, and lists some important recurrence relations and examples. The document is intended as a short survey of essential properties of these important mathematical functions.

1. SPECIAL FUNCTIONS and POLYNOMIALS
Gerard 't Hooft
Stefan Nobbenhuis
Institute for Theoretical Physics
Utrecht University, Leuvenlaan 4
3584 CC Utrecht, the Netherlands
and
Spinoza Institute
Postbox 80.195
3508 TD Utrecht, the Netherlands
Many of the special functions and polynomials are constructed along standard
procedures In this short survey we list the most essential ones.
April 8, 2013
1

2. 1 Legendre Polynomials P`(x) .
Dierential Equation:
(1 x2) P`
00(x) 2x P`
0(x) + `(` + 1) P`(x) = 0 ;
or
d
dx
(1 x2)
d
dx
P`(x) + `(` + 1) P`(x) = 0 : (1.1)
Generating function:
1X
`=0
P`(x)t` = (1 2xt + t2)1
2 for jtj 1; jxj 1: (1.2)
Orthonormality:
Z 1
-1
P`(x) P` 0(x) dx =
2
2` + 1
` ` 0 ; (1.3)
1X
`=0
P`(x)P`(x0)(2` + 1) = 2(x x0) : (1.4)
Expressions forP`(x) :
P`(x) =
1
2`
[X`=2]
=0
(1) (2` 2)!
! (` )! (` 2)!
x`2 (1.5)
=
1
`! 2`
d
dx
`
(x2 1)` ; (1.6)
=
1
Z
0
(x +
p
x2 1 cos ')` d' : (1.7)
Recurrence relations:
` P`1 (2` + 1) x P` + (` + 1) P`+1 = 0 ;
P` = x P`1 +
x2 1
`
P 0
`1 ;
xP 0
` ` P` = P 0
`1 ;
xP 0
` + (` + 1) P` = P 0
`+1 ;
d
dx
[P`+1 P`1] = (2` + 1) P`: (1.8)
Examples:
2(3x2 1) ; P3 = 1
2x(5x2 3) : (1.9)
P0 = 1 ; P1 = x ; P2 = 1
1

3. 2 Associated Legendre Functions Pm
` (x) .
Dierential equation:
(1 x2) Pm
` (x) 00 2x Pm
` (x) 0 +
`(` + 1)
m2
1 x2
Pm
` (x) = 0 : (2.1)
Generating function:
1X
`=0
X`
m=0
Pm
` (x) zm y`
m!
=
h
1 2y
x + z
p
1 x2
+ y2
i
1
2 : (2.2)
Orthogonality:
Z 1
-1
Pm
` (x) Pm
` 0 (x) dx =
2
2` + 1
(` + m)!
(` m)!
` ` 0 ; ( `; ` 0 m) : (2.3)
1X
(2` + 1)
`=m
(` m)!
(` + m)!
Pm
` (x) Pm
` (x0) = 2(x x0) ; ( jxj 1 and jx0j 1 ) : (2.4)
Expressions for Pm
` (x)1:
Pm
` (x) = (1 x2)
1
2m
d
dx
!m
P`(x) : (2.5)
Pm
` (x) =
(` + m)!
`!
(1)m=2
Z
0
x +
p
x2 1 cos '
`
cosm'd' : (2.6)
Recurrence relations:
Pm+1
`
2mx
p
1 x2
Pm
` + f`(` + 1) m(m 1)gPm1
` = 0 (2.7)
p
1 x2 Pm+1
` (x) = (1 x2) Pm
` (x) 0 + mxPm
` (x) ;
(2` + 1)x Pm
` = (` + m) Pm
`1 + (` + 1 m) Pm
`+1 ; (2.8)
x Pm
p
1 x2 Pm1
` = Pm
`1 (` + 1 m)
` ;
Pm
`+1 Pm
`1 = (2` + 1) Pm1
`
p
1 x2 ; (2.9)
and various others.
Examples:
P1
1 =
p
1 x2 ; P2
2 = 3(1 x2) ;
P1
2 = 3x
p
1 x2 ; P2
3 = 15 x(1 x2) : (2.10)
1Note that some authors de

4. ne Pm
` (x) with a factor (1)m, giving Pm
` (x) = (1)m(1
x2)
1
2m
d
dx
m
P`(x) . Obviously this minus sign propagates to the generating function, the recurrence
relations and the explicit examples, when m is odd.
2

5. 3 Bessel Jn(x) and Hankel Hn(x) functions.
Dierential equation (for both Jn and Hn ):
x2 J00
n(x) + x J0
n(x) + (x2 n2) Jn(x) = 0 : (3.1)
Generating function (if n integer):
1X
n=1
Jn(x)
s
n
= e
x
2 (s 2
s ) ; (3.2)
Jn = (1)n Jn :
Orthogonality:
Z 1
0
Jn() Jn(

6. ) d =
1
(jj j

7. j) : (3.3)
Z a
0
Jn() Jn(

8. ) d =
a2
2
fJn+1(a)g2

9. : (3.4)
if in the 2nd relation ;

10. are roots of the equation Jn() = 0.
Expressions for Jn(x) (for n integer):
Jn(x) =
1X
k=0
(1)k
k! (k + n)!
x
2
n+2k
=
1
2i
x
2
n
I
tn1dt e tx2=4t : (3.5)
Jn(x) =
1
Z
0
cos (n x sin ) d : (3.6)
Recurrence relations (for both Jn and Hn ):
d
dx
fxn Jn(x)g = xn Jn1(x) ;
Jn1(x) + Jn+1(x) =
2n
x
Jn(x) ;
J 0
n(x) = Jn1(x)
n
x
Jn(x) =
n
x
Jn(x) Jn+1(x) = 1
2 (Jn1(x) Jn+1(x)) : (3.7)
Relations between Hankel and Bessel functions:
H(1)
n (x) =
i
sin n
eniJn(x) Jn(x)
;
H(2)
n (x) =
i
sin n
eniJn(x) Jn(x)
; (3.8)
so that
Jn(x) = 1
2
H(1)
;
n (x) + H(2)
n (x)
Jn(x) = 1
2
eniH(1)
: (3.9)
n (x) + eniH(2)
n (x)
3

11. 4 Spherical Bessel Functions j`(x).
Dierential equation:
(xj`)00 +
x
`(` + 1)
x
j` = 0 : (4.1)
Generating Function:
1X
`=0
j`(x) t`
`!
= j0
p
x2 2xt
: (4.2)
Orthogonality:
Z 1
0
x2j`(x) j`(