Following on from my previous post on infinite products, here are some more to enjoy! Like last time, proof outlines are at the end.
In particular x=2 gives von Seidel‘s product:
(6) for |x| < 1
(in each fraction the numerator is an odd prime, the denominator is the nearest multiple of 4 to the numerator)
(in each fraction the numerator is an odd prime, the denominator is the nearest even non-multiple of 4 to the numerator)
(in each fraction the numerator and denominator differ by 1, sum to the odd primes, and numerator is even)
(in each fraction the numerator and denominator differ by 1, sum to odd numbers that are not prime, and numerator is even)
(18) Pentagonal number theorem:
A. Products based on geometric series
The following proof of (1) is based on . We start with repeated application of the difference of perfect squares identity:
In this expression let y=1. After rearrangement this gives
Next we take the limit of both sides as . The left side becomes and so we have
As noted in  one can also set in (*) and arrive at
so letting gives us (2). It need not be arrived at via the double angle formula!
Setting in (2) automatically gives (3), which also can be derived by considering regular (2^n)-gons being approximated by a circle in the limit.
Next (4) is based on the infinite series while (5) is apparent from the similar equality
Finally, (6) is based on the fact that every positive integer has a unique representation in binary, so every term is represented in exactly one product on the left side (e.g. ), and so
which is summed as a geometric series for |x| < 1.
B. Euler products
The next products over the primes [(7)-(14)] are due to Euler and come from his expansion of the Riemann zeta function in (7):
This is a essentially a statement of unique factorisation, that every positive integer n can be written uniquely as a product of primes.
Setting s=2 gives (8). Setting s=4 gives
The left side is equal to .
One way of seeing this is by taking the term in the infinite product expansion (seen in my previous post). This gives
Finally one obtains (9) by dividing (**) by (8). (10) simply arises from dividing (8) by (9).
To prove (11) we proceed in a similar manner to proving (7), this time using unique factorisation of the odd natural numbers, and that an odd number that is 3 modulo 4 must have all prime factors that are 3 modulo 4 occurring an odd number of times:
To be more precise, if m is the largest prime less than N, one can show  that the difference
Setting s to 1 and using the sum gives (11). We obtain (12) by dividing 3/4 times (8) by (11).
Dividing 3/4 times (8) by the square of (11) leads to (13) after some cancellation.
Dividing this by (13) leads to (14).
C. Formulas involving e
We obtain (14) by using the following infinite sum form of e:
We can see by the definition of that the product . Then it can be seen that is the numerator of the partial sum . The result follows by taking the limit .
To obtain (15) we use Stirling’s formula for the asymptotic form of the factorial function:
Next we observe that each term in the right side can be written in terms of factorials and powers of 2. For example,
The general term is of the form
When these terms are multiplied, the product telescopes and we end up with
We remark that the product converges rapidly to e, since even if we use the more precise form and replace with the Wallis product expansion, the left and right sides of (15) can be shown to be similar for N large.
It is amazing to see that (15) becomes Wallis’s product (***) for when the exponents are set to 1 (while keeping the same fractions)!
(16) can be easily obtained from (15) by multiplying both sides by
(although Catalan’s result, obtained through more involved means, long preceded that of Pippenger).
More similar products are given in .
D. Pentagonal number theorem
This result, again due to Euler, shows how much cancellation there is when the product is expanded. The coefficient of is the number of partitions of n into an even number of parts minus the number of partitions of n into an odd number of parts. The theorem says that this number is zero unless n is a pentagonal number (i.e. of the form k(3k-1)/2), in which case it is 1 or -1. Refer to this Wikipedia entry of this theorem for a combinatorial proof.
 T.J. Osler, Interesting finite and infinite products from simple algebraic identities, The Mathematical Gazette, 90(2006), pp. 90-93. Available here.
 K. Brown, Infinite Products and a Tangent Fan. Mathpages link
 P. Loya, Amazing and Aesthetic Aspects of Analysis: On the incredible infinite, available at http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdf
 J. Sondow and H. Yi, New Wallis- and Catalan-Type Infinite Products for π , e, and sqrt(2+ sqrt(2)), available at http://arxiv.org/ftp/arxiv/papers/1005/1005.2712.pdf