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:
“I have always suspected that too much knowledge is a dangerous thing. It is a boon to people who don’t have deep feelings; their pleasure comes from what they know about things, and their pride from showing off what they know. But this only emphasizes the difference between the artist and the scholar.”
—Margaret Anderson (1886–1973)
“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)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)
“The experience and behaviour that gets labelled schizophrenic is a special strategy that a person invents in order to live in an unlivable situation.”
—R.D. (Ronald David)
“Science has nothing to be ashamed of even in the ruins of Nagasaki. The shame is theirs who appeal to other values than the human imaginative values which science has evolved.”
—Jacob Bronowski (1908–1974)