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If, for n>=0, beta_n=sum_(r=0)^n(alpha_r)/((q;q)_(n-r)(aq;q)_(n+r)), (1) then beta_n^'=sum_(r=0)^n(alpha_r^')/((q;q)_(n-r)(aq;q)_(n+r)), (2) where alpha_r^' = ...

A transformation formula for continued fractions (Lorentzen and Waadeland 1992) which can, for example, be used to prove identities such as ...

A division algebra, also called a "division ring" or "skew field," is a ring in which every nonzero element has a multiplicative inverse, but multiplication is not ...

There are many unsolved problems in mathematics. Several famous problems which have recently been solved include: 1. The Pólya conjecture (disproven by Haselgrove 1958, ...

The distinct prime factors of a positive integer n>=2 are defined as the omega(n) numbers p_1, ..., p_(omega(n)) in the prime factorization ...

The Jacobi triple product is the beautiful identity product_(n=1)^infty(1-x^(2n))(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2))=sum_(m=-infty)^inftyx^(m^2)z^(2m). (1) In terms of the ...

A q-series is series involving coefficients of the form (a;q)_n = product_(k=0)^(n-1)(1-aq^k) (1) = product_(k=0)^(infty)((1-aq^k))/((1-aq^(k+n))) (2) = ...

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep ...

The lemniscate functions arise in rectifying the arc length of the lemniscate. The lemniscate functions were first studied by Jakob Bernoulli and Giulio Fagnano. A historical ...

Thomae's theorem, also called Thomae's transformation, is the generalized hypergeometric function identity (1) where Gamma(z) is the gamma function, _3F_2(a,b,c;e,f;z) is a ...