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_3F_2[n,-x,-y; x+n+1,y+n+1] =Gamma(x+n+1)Gamma(y+n+1)Gamma(1/2n+1)Gamma(x+y+1/2n+1) ×Gamma(n+1)Gamma(x+y+n+1)Gamma(x+1/2n+1)Gamma(y+1/2n+1), (1) where _3F_2(a,b,c;d,e;z) is a ...

The factorial n! is defined for a positive integer n as n!=n(n-1)...2·1. (1) So, for example, 4!=4·3·2·1=24. An older notation for the factorial was written (Mellin 1909; ...

A mathematical statement that one quantity is greater than or less than another. "a is less than b" is denoted a<b, and "a is greater than b" is denoted a>b. "a is less than ...

The Legendre symbol is a number theoretic function (a/p) which is defined to be equal to +/-1 depending on whether a is a quadratic residue modulo p. The definition is ...

A type of integral containing gamma functions in its integrand. A typical such integral is given by ...

The transform inverting the sequence g(n)=sum_(d|n)f(d) (1) into f(n)=sum_(d|n)mu(d)g(n/d), (2) where the sums are over all possible integers d that divide n and mu(d) is the ...

A game which is played by the following rules. Given one or more piles (nim-heaps), players alternate by taking all or some of the counters in a single heap. The player ...

product_(k=1)^(infty)(1-x^k) = sum_(k=-infty)^(infty)(-1)^kx^(k(3k+1)/2) (1) = 1+sum_(k=1)^(infty)(-1)^k[x^(k(3k-1)/2)+x^(k(3k+1)/2)] (2) = (x)_infty (3) = ...

Legendre showed that there is no rational algebraic function which always gives primes. In 1752, Goldbach showed that no polynomial with integer coefficients can give a prime ...

Let pi_(m,n)(x) denote the number of primes <=x which are congruent to n modulo m (i.e., the modular prime counting function). Then one might expect that ...