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A q-analog of the Chu-Vandermonde identity given by where _2phi_1(a,b;c;q,z) is the q-hypergeometric function. The identity can also be written as ...

A hypergeometric identity discovered by Ramanujan around 1910. From Hardy (1999, pp. 13 and 102-103), (1) where a^((n))=a(a+1)...(a+n-1) (2) is the rising factorial (a.k.a. ...

For P, Q, R, and S polynomials in n variables [P·Q,R·S]=sum_(i_1,...,i_n>=0)A/(i_1!...i_n!), (1) where A=[R^((i_1,...,i_n))(D_1,...,D_n)Q(x_1,...,x_n) ...

sum_(k=0)^m(phi_k(x)phi_k(y))/(gamma_k)=(phi_(m+1)(x)phi_m(y)-phi_m(x)phi_(m+1)(y))/(a_mgamma_m(x-y),) (1) where phi_k(x) are orthogonal polynomials with weighting function ...

The identity F_n^4-F_(n-2)F_(n-1)F_(n+1)F_(n+2)=1, where F_n is a Fibonacci number.

This is proven in Rademacher and Toeplitz (1957).

sum_(n=0)^(infty)(-1)^n[((2n-1)!!)/((2n)!!)]^3 = 1-(1/2)^3+((1·3)/(2·4))^3+... (1) = _3F_2(1/2,1/2,1/2; 1,1;-1) (2) = [_2F_1(1/4,1/4; 1;-1)]^2 (3) = ...

The vector triple product identity Ax(BxC)=B(A·C)-C(A·B). This identity can be generalized to n dimensions,

Given the generating functions defined by (1+53x+9x^2)/(1-82x-82x^2+x^3) = sum_(n=1)^(infty)a_nx^n (1) (2-26x-12x^2)/(1-82x-82x^2+x^3) = sum_(n=0)^(infty)b_nx^n (2) ...

e^(i(ntheta))=(e^(itheta))^n. (1) From the Euler formula it follows that cos(ntheta)+isin(ntheta)=(costheta+isintheta)^n. (2) A similar identity holds for the hyperbolic ...