Difference Equation, Functional Equation and Special Values
The Barnes G-function satisfies the difference equation
with normalisation G(1) = 1. The difference equation implies that G takes the following values at integer arguments:
and thus
where Γ denotes the Gamma function and K denotes the K-function. The difference equation uniquely defines the G function if the convexity condition: is added.
The difference equation for the G function and the functional equation for the Gamma function yield the following functional equation for the G function, originally proved by Hermann Kinkelin:
Read more about this topic: Barnes G-function
Famous quotes containing the words difference, functional, equation, special and/or values:
“It don’t make no difference whether you do right or wrong, a person’s conscience ain’t got no sense, and just goes for him anyway. If I had a yaller dog that didn’t know no more than a person’s conscience does, I would pison him.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Indigenous to Minnesota, and almost completely ignored by its people, are the stark, unornamented, functional clusters of concrete—Minnesota’s grain elevators. These may be said to express unconsciously all the principles of modernism, being built for use only, with little regard for the tenets of esthetic design.”
—Federal Writers’ Project Of The Wor, U.S. public relief program (1935-1943)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)
“The line that I am urging as today’s conventional wisdom is not a denial of consciousness. It is often called, with more reason, a repudiation of mind. It is indeed a repudiation of mind as a second substance, over and above body. It can be described less harshly as an identification of mind with some of the faculties, states, and activities of the body. Mental states and events are a special subclass of the states and events of the human or animal body.”
—Willard Van Orman Quine (b. 1908)
“Normality highly values its normal man. It educates children to lose themselves and to become absurd, and thus to be normal. Normal men have killed perhaps 100,000,000 of their fellow normal men in the last fifty years.”
—R.D. (Ronald David)