In mathematics, the indefinite product operator is the inverse operator of Q ( f ( x ) ) = f ( x + 1 ) f ( x ) . It is like a discrete version of the indefinite product integral. Some authors use term discrete multiplicative integration
Thus
Q ( ∏ x f ( x ) ) = f ( x ) . More explicitly, if ∏ x f ( x ) = F ( x ) , then
F ( x + 1 ) F ( x ) = f ( x ) . If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore each indefinite product actually represents a family of functions, differing by a multiplicative constant.
If T is a period of function f ( x ) then
∏ x f ( T x ) = C f ( T x ) x − 1 Indefinite product can be expressed in terms of indefinite sum:
∏ x f ( x ) = exp ( ∑ x ln f ( x ) ) Some authors use the phrase "indefinite product" in a slightly different but related way to describe a product in which the numerical value of the upper limit is not given. e.g.
∏ k = 1 n f ( k ) .
∏ x f ( x ) g ( x ) = ∏ x f ( x ) ∏ x g ( x ) ∏ x f ( x ) a = ( ∏ x f ( x ) ) a ∏ x a f ( x ) = a ∑ x f ( x ) This is a list of indefinite products ∏ x f ( x ) . Not all functions have an indefinite product which can be expressed in elementary functions.
∏ x a = C a x ∏ x x = C Γ ( x ) ∏ x x + 1 x = C x ∏ x x + a x = C Γ ( x + a ) Γ ( x ) ∏ x x a = C Γ ( x ) a ∏ x a x = C a x Γ ( x ) ∏ x a x = C a x 2 ( x − 1 ) ∏ x a 1 x = C a Γ ′ ( x ) Γ ( x ) ∏ x x x = C e ζ ′ ( − 1 , x ) − ζ ′ ( − 1 ) = C e ψ ( − 2 ) ( z ) + z 2 − z 2 − z 2 ln ( 2 π ) = C K ( x ) (see
K-function)
∏ x Γ ( x ) = C Γ ( x ) x − 1 K ( x ) = C Γ ( x ) x − 1 e z 2 ln ( 2 π ) − z 2 − z 2 − ψ ( − 2 ) ( z ) = C G ( x ) (see Barnes G-function)
∏ x sexp a ( x ) = C ( sexp a ( x ) ) ′ sexp a ( x ) ( ln a ) x (see super-exponential function)
∏ x x + a = C Γ ( x + a ) ∏ x a x + b = C a x Γ ( x + b a ) ∏ x a x 2 + b x = C a x Γ ( x ) Γ ( x + b a ) ∏ x x 2 + 1 = C Γ ( x − i ) Γ ( x + i ) ∏ x x + 1 x = C Γ ( x − i ) Γ ( x + i ) Γ ( x ) ∏ x csc x sin ( x + 1 ) = C sin x ∏ x sec x cos ( x + 1 ) = C cos x ∏ x cot x tan ( x + 1 ) = C tan x ∏ x tan x cot ( x + 1 ) = C cot x 