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> Line 47: Line 47: Paul Berry, ''[:../Bibliography#SAPL_Ref:SHARP APL Reference Manual]'', I.P. Sharp Associates, 1979-03, page 132. Paul Berry, ''[[../Bibliography#SAPL_Ref|SHARP APL Reference Manual]]'', I.P. Sharp Associates, 1979-03, page 132.

The (complete) beta function is defined by

`(x beta y) = (gamma x) * (gamma y) % gamma x+y`

Since (gamma x)=!x-1 and (x!y)=(!y)%(!x)*!y-x , we have:

```x beta y
(gamma x) * (gamma y) % gamma x+y
(!x-1) * (!y-1) % !x+y-1
((!x-1) * (!y) % !x+y-1) % y
(% (x-1)!x+y-1) % y
% y * (x-1)!x+y-1```

Therefore:

`beta=: ] %@* [ !&<: +`

```   0.5 beta 17 17.5
0.433055 0.426734

beta"0/~ >: i.10x
1   1r2   1r3    1r4     1r5     1r6     1r7      1r8      1r9     1r10
1r2   1r6  1r12   1r20    1r30    1r42    1r56     1r72     1r90    1r110
1r3  1r12  1r30   1r60   1r105   1r168   1r252    1r360    1r495    1r660
1r4  1r20  1r60  1r140   1r280   1r504   1r840   1r1320   1r1980   1r2860
1r5  1r30 1r105  1r280   1r630  1r1260  1r2310   1r3960   1r6435  1r10010
1r6  1r42 1r168  1r504  1r1260  1r2772  1r5544  1r10296  1r18018  1r30030
1r7  1r56 1r252  1r840  1r2310  1r5544 1r12012  1r24024  1r45045  1r80080
1r8  1r72 1r360 1r1320  1r3960 1r10296 1r24024  1r51480 1r102960 1r194480
1r9  1r90 1r495 1r1980  1r6435 1r18018 1r45045 1r102960 1r218790 1r437580
1r10 1r110 1r660 1r2860 1r10010 1r30030 1r80080 1r194480 1r437580 1r923780```

The (x beta y) = %y*(x-1)!x+y-1 identity can be found on Paul Berry, SHARP APL Reference Manual, I.P. Sharp Associates, 1979-03, page 132.

Essays/Beta Function (last edited 2008-12-08 10:45:29 by anonymous)