One of Wallis formulas is
for
. This formula can be proved by various methods [1] [2] [3] [4] including a repeated application of a reduction formula such as
. Note that
and
are coordinates
of a point on the unit sphere in R2. Since the above formula involves an integration over the unit circle in R2, its extension to higher dimensions is of interest.
For each
, let
be its Euclidean norm. We call
, where
are non-negative integers, a multi-index, and
its degree. Set
and
. Let
be the unit sphere in Rn and
be its surface measure. Let
stand for the ball of radius r centered at a. The gamma function is defined as
, for
. The generalized Wallis’s formula is a special case of the following theorem.
Theorem 1 (i)
, if any
is odd. In particular, the integral equals zero if
is odd.
(ii)
.
Setting
and
for
in the theorem, the generalized Wallis’s formula follows
Note that for
, (ii) is equivalent to the well-known formula
(1)
where
is the surface area of the unit sphere in Rn. Theorem 1 is interesting in its own right and has further applications. For example, for a polynomial
of degree m, one may express
as a simple
polynomial of degree
in r. In the following we use polar coordinates
.
Here
as given by (ii), and [.] is the bracket function.
Proof of Theorem 1. (i) The proof is by induction on
.
If
then
for some i. Therefore,
by the symmetry of the sphere.
Assume now the assertion is true for
for some
. Let
and assume, without loss of generality, that
is odd. Applying the divergence theorem results in
(2)
If
, the last integral in (2) is zero. Otherwise, a conversion to polar coordinates in (2), yields,
where
. The last integral is now zero, by the induction hypothesis.
ii) The proof is by induction on
.
For
, we must establish (1). Let
. Writing
as a product of integrals and using polar coordinates in R2 followed by a change of variables, one obtains
We used a change of variable
in the previous integral. Converting to polar coordinates for Rn results in
Identity (1) follows immediately from the last equation.
Now suppose the claim is true for
. Let
. We may assume, without loss of generality, that
. Applying the divergence theorem followed by a conversion to polar coordinates leads to
where
. Since
, and using the fact that
along with the induction hypothesis, we get