One of Wallis formulas is
. This formula can be proved by various methods     including a repeated application of a reduction formula such as
. Note that
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.
be its Euclidean norm. We call
are non-negative integers, a multi-index, and
its degree. Set
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
. 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
in the theorem, the generalized Wallis’s formula follows
Note that for
, (ii) is equivalent to the well-known formula
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
as given by (ii), and [.] is the bracket function.
Proof of Theorem 1. (i) The proof is by induction on
for some i. Therefore,
by the symmetry of the sphere.
Assume now the assertion is true for
and assume, without loss of generality, that
is odd. Applying the divergence theorem results in
, the last integral in (2) is zero. Otherwise, a conversion to polar coordinates in (2), yields,
. The last integral is now zero, by the induction hypothesis.
ii) The proof is by induction on
, we must establish (1). Let
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
. We may assume, without loss of generality, that
. Applying the divergence theorem followed by a conversion to polar coordinates leads to
, and using the fact that
along with the induction hypothesis, we get