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and the corresponding duplication formula for f is f ( 2 x ) = g ( f ( x ) ).
As we all know, the duplication formula for the gamma function is given by Gamma (z Gamma biggl z + frac{1}{2} biggr) = {2^{1 - 2z}} sqrt{pi }Gamma (2z).
and the duplication formula for the classical Hurwitz zeta function ζ ( s, x ) + ζ ( s, x + 1 2 ) = 2 s ζ ( s, 2 x ), (2.10).
As is well known, the duplication formula for the gamma function is given by Γ ( z ) Γ ( z + 1 2 ) = 2 1 − 2 z π Γ ( 2 z ).
In particular, the case a = 2 in (2.7) gives the duplication formula for the q-zeta function ζ q 2 ( s, x ) + q ζ q 2 ( s, x + 1 2 ) = [ 2 ] q s ζ q ( s, 2 x ).
In fact, we establish some integral formulas for the Bessel function J ν ( z ) expressed in terms of the generalized hypergeometric function F q p. To do this, we recall the well-known Legendre duplication formula for the Gamma function Γ: π Γ ( 2 z ) = 2 2 z − 1 Γ ( z ) Γ ( z + 1 2 ) ( z ≠ 0, − 1 2, − 1, − 3 2, … ), (2.6).
Apologies for duplication!
How to check for duplication.
