The Wallis formula follows from the infinite product representation of the sine
(1)
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Taking gives
(2)
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so
(3)
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(4)
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An accelerated product is given by
(5)
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(6)
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where
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(Guillera and Sondow 2005, Sondow 2005). This is analogous to the products
(8)
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and
(9)
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(Sondow 2005).
A derivation of equation (◇) due to Y. L. Yung (pers. comm., 1996; modified by J. Sondow, pers. comm., 2002) defines
(10)
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(11)
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(12)
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where is a polylogarithm and is the Riemann zeta function, which converges for . Taking the derivative of (11) gives
(13)
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which also converges for , and plugging in then gives
(14)
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(15)
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(16)
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Now, taking the derivative of the zeta function expression (◇) gives
(17)
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and again setting yields
(18)
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(19)
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(20)
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(21)
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where
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(OEIS A075700) follows from the Hadamard product for the Riemann zeta function. Equating and squaring (◇) and (◇) then gives the Wallis formula.
This derivation of the Wallis formula from using the Hadamard product can also be reversed to derive from the Wallis formula without using the Hadamard product (Sondow 1994).
The Wallis formula can also be expressed as
(23)
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The q-analog of the Wallis formula with is
(24)
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(25)
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(OEIS A065446; Finch 2003), where is the q-Pochhammer symbol. This constant is , where is the constant encountered in digital tree searching. The form of the product is exactly the generating function for the partition function P due to Euler, and is related to q-pi.