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Usage:
polygamma(x [,n]), x REAL with positive elements or a structure with
REAL components with positive elements, integer n >= 0
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Keywords:
transformations
polygamma(x,0) and polygamma(x) both return the digamma function (first
derivative of log(gamma(x))) of the elements of x, when x is a REAL
scalar, vector, matrix or array with positive elements. The result has
the same shape as x. You can use digamma(x) instead.
polygamma(x, n), where n > 0 is an integer returns the n-th derivative
of the digamma function ((n+1)-th derivative of log(gamma(x))).
polygamma(x, n, scale:T) returns (-1)^(n+1)*n!*polygamma(x,n).
For n >= 1, polygamma(x, n, scale:T) = sum((x+k)^(-n-1),k=0,1,2,...,oo).
In particular, polygamma(1,n,scale:T) computes zeta(n+1), where zeta(s)
is the Riemann Zeta function.
When x is a structure, all of whose non-structure components are REAL
with positive elements, polygamma(x [,n] [,scale:T]) returns a structure
of the same shape and with the same component names as x with each
non-structure component transformed by polygamma().
polygamma(x, n) can also be used when x is a CHARACTER variable and n,
if present, is a quoted string or CHARACTER scalar or REAL scalar. The
result is a CHARACTER variable of the same shape as x describing the
transformation. See example below.
Any element of x that is "" or starts with '@', '(', '[', '{', '<', '/'
or '\' is not modified. This can be useful for creating labels for a
transformed variable.
Examples:
Cmd> polygamma(run(10)) # or polygamma(run(10),0), or digamma(run(10))
(1) -0.57722 0.42278 0.92278 1.2561 1.5061
(6) 1.7061 1.8728 2.0156 2.1406 2.2518
Cmd> polygamma(run(1,2,.25),1) # trigamma
(1) 1.6449 1.1973 0.9348 0.7641 0.64493
Cmd> polygamma(vector("x","y"),3) # or polygamma(vector("x","y"),"3")
(1) "polygamma(x,3)"
(2) "polygamma(y,3)"
Cmd> print(nsig:17,polygamma(1,23,scale:T),name:"zeta(24)")
zeta(24):
(1) 1.0000000596081891
See also digamma(), lgamma(), 'transformations'.
Gary Oehlert
2003-01-15