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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
Chapter 3 0
Special Functions
and Their Properties
Throughout Chapter 30 it is assumed that
n
is a positive integer unless otherwise spec-
ified.
30.1 Some Coefficients, Symbols, and Numbers
30.1.1 Bin omial Coefficients
Definitions.
C
k
n
=
n
k
=
n!
k! (n k)!
, where k = 1, . . . , n;
C
0
a
= 1, C
k
a
=
a
k
= (1)
k
(a)
k
k!
=
a(a 1) . . . (a k + 1)
k!
, where k = 1, 2, . . .
Here a is an arbitrary real number.
Generalization. Some properties.
General case:
C
b
a
=
Γ(a + 1)
Γ(b + 1)Γ(a b + 1)
, where Γ(x) is the gamma function.
Properties:
C
0
a
= 1, C
k
n
= 0 for k = 1, 2, . . . or k > n,
C
b+1
a
=
a
b + 1
C
b
a1
=
a b
b + 1
C
b
a
, C
b
a
+ C
b+1
a
= C
b+1
a+1
,
1509
1510 SPECIAL FUNCTIONS AND THEIR PR
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Publisher Resources

ISBN: 9781466581494