Concrete mathematics : a foundation for computer science

5.5 HYPERGEOMETRIC FUNCTIONS 205
Anything that has
The study of hypergeometric series was launched many years ago by Euler,
Gauss, and Riemann; such series, in fact, are still the subject of considerable
research. But hypergeometrics have a somewhat formidable notation,
which takes a little time to get used to.
survived for centuries
with such
awesome notation
The general hypergeometric series is a power series in z with m + n
parameters, and it is defined as follows in terms of rising factorial powers:
must be really
useful.
i; i; k
al, ..', aIlI
F a’ ...am 4.
( bl, .-.,bn 1) ’ = k>O 5 by. . . bi k!
(5.76)
To avoid division by zero, none of the b’s may be zero or a negative integer.
Other than that, the a’s and b’s may be anything we like. The notation
‘F(al,. . . ,a,,,; bl,. . . , b,; z)’ is also used as an alternative to the two-line form
(5.76), since a one-line form sometimes works better typographically. The a’s
are said to be upper parameters; they occur in the numerator of the terms
of F. The b’s are lower parameters, and they occur in the denominator. The
final quantity z is called the argument.
Standard reference books often use ’ ,,,F,’ instead of ‘F’ as the name of a
hypergeometric with m upper parameters and n lower parameters. But the
extra subscripts tend to clutter up the formulas and waste our time, if we’re
compelled to write them over and over. We can count how many parameters
there are, so we usually don’t need extra additional unnecessary redundancy.
Many important functions occur as special cases of the general hypergeometric;
indeed, that’s why hypergeometrics are so powerful. For example, the
simplest case occurs when m = n = 0: There are no parameters at all, and
we get the familiar series
F ( 1~) = &$ = e’.
Actually the notation looks a bit unsettling when m or n is zero. We can add
an extra ‘1’ above and below in order to avoid this:
In general we don’t change the function if we cancel a parameter that occurs
in both numerator and denominator, or if we insert two identical parameters.
The next simplest case has m = 1, al = 1, and n = 0; we change the
parameterstom=2, al =al=l, n=l,andbl =l,sothatn>O. This
series also turns out to be familiar, because 1’ = k!: