Short Communication

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Texturesand Microstructures,1988,Vol. 10,pp.37-40

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Short

Communication

H. BAVINCK and R. F. SWARTTOUW

DelftUniversityof Technology,DepartmentofTechnical Mathematics

andInformatics,Julianalaan 132,2628BLDelft,The Netherlands

Aconjecture byBrakman(1987)concerninga suminvolvingbinomial coefficients is

proved by usingformulae forhypergeometricfunctions.

KEY WORDS: Hypergeometric functions,binomial coefficients.

In a recentpaperBrakman (1987)conjectured that

(l+m-2)(

l-m+2

)

1-n _k _n-m+k+2 n-m+k+l (1+m 2)!(1-n+1)!(n m+1)! (20! x l-n+l l-n+ for 1,m, n

,

m<-n<-1. Wepresentaproofofthisformula, using hypergeometricfunctions.

Thehypergeometricfunctions _2F and _3F2are definedby

and

2Fl(

a’ b c (a)k(b)k .k Z Z =o (c)kt

z)

",

(a)k(b)k(C)k k=O (d)k(e)kk!

zk’

where (a)k=a(a +1)... (a+k-1).

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38 SHORT COMMUNICATION

Weshall onlyusethecasewhere a, b,c, d,e Zand z 1. For a positive integer n we maywrite

and

(n)k

(n

+k-1)! (n- 1)!

n

(--n)k=(--1)k(n_k)----’----

7:

for k<-n, (--n)k=O for k>n.

E

(-1)k k=O k n-m+k+2

(

21

)

n-m+k+l 1--tl

E

(-1)k k=O (1+rn 2)!(1-rn+2)!(n rn+k+1)! (2/+m n k 1)! k!(l+m k 2)!(n m+k+2)!(1-n k)!(21)! (l m +2)!(n m+1)!(21+rn n+1)! (21)!(n rn

₊

2)!(1- n)! ’-" (n 0,(2- m),(n m+2), x ,=o (n m 21+1),(n m+3)k (l-m+2)t(n m+1)!(2/+m n 1)! (21)!(n rn+2)!(1- n)!

3F2[n-1,2-1-m,n-m+2

X

kn

-m-21+1, n-m +3

Wenow use theformula (Luke, 1969,p. 111, formula(38))

3F2

d+l,c+l z c-d2F\d+l z

d _./a,b,c

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SHORT COMMUNICATION 39 towrite

3F2f

n-1, 2-1-m, n m+2 -m-21/2,n -m +3 n-m+2 2/+1 -m-21+2

1)

-n-m-21+21+1

1

nn-l,

2-l-m, n-m +2 x_3F2 -m-21+1,n-m +3 or

3F2(n-

l,2-l- m, n m+2 \n-m-21+1, n-m +3

1)

21+m-n-13F2 m 21+2, n m

₊

3 n-m+2 21+m-n-1

2F(-1,2-1-m -m-2/+2

The hypergeometric functions on the right can be evaluated by

using the formulae

2F(-n,

b.

₁₎

(c-b) (Luke, 1969, p.99formula(3)) c and -n, a, b

F

_a

₊

_b_{-n -c}₊_l, _c Weobtain (c a),,(c b),, 1

=(c),,(c-a-b),,

(Luke, 1969, p. 103 formula(2)).

3F2(nn

l’2-l- m’ n m+2

)

-m-21+l,n-m+3 1 (2/+1)(n

+

l+1)/-n(1)t-n (2/+m n 1)(n m+3)/_(/+m 1)l_n (n m

₊

2)(n l)l-,, (21

+

rn n 1)(n rn 2l

+

2)l-n

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40 SHORTCOMMUNICATION (2l+1)(/-n)!(21)!(l+m 2)!(n m +2)! Hence (2l+rn n 1)(1+n)!(l rn+2)!(21+rn n _2)! (n m

+

2)(/- n)!(l+rn 2)! (2l+rn n 1)(21+rn n 2)! (l- n)t(/+rn 2)!(n rn+2)! (2/+rn n 1)!

[

(2/+1), 1

]

(l+n)t(l rn+

2)!-

(n rn+1)!

E

(-I):

k=O k n-m+k+2 n-m+k+l (1+rn 2)!(n rn+1)t(l-n+ (20!

[

(2l+1), (l-m +2),

]

(1+n)!(l n+

1)!-

(1 n+1)!(n m+1)! (1+m 2)!(n m+1)!(1 n+1)! (21)t

[(

21+1

)_(1-m+12)]

l-n+1 l-n+

which isthe desiredresult.

References

Brakman,C. M.(1987).Evaluationof the integral_fl Pr’n(X)P’--2"n(x)dx. Textures

and Microstructures,7,207-210.

Luke, Y. L. (1969). The specialfunctions and their approximations, volume I,