A new type of weighted quadrature rules and its relation with orthogonal polynomials

In this research, we introduce a new type of weighted quadrature rules as∫αβρ(x)(f(x)-Pm-1(x;f))dx=∑i=1nai,mf(m)(bi,m)+Rn(m)[f],in which Pm-1(x;f)=∑j=0m-1f(j)(λ)(x-λ)j/j!; λ ∈ R; m ∈ N; ρ(x) is a positive function; f(m) (x) denotes the mth derivative of the function f(x  ) and Rn(m)[f] is the error function. We determine the error function analytically and obtain the unknowns {ai,m,bi,m}i=1n explicitly so that the above formula is exact for all polynomials of degree at most 2n + m − 1. In particular, we emphasize on the sub-case∫αβρ(x)(f(x)-f(λ))dx=∑i=1nai,1f′(bi,1)+Rn(1)[f],with the precision 2n (one degree higher than Gauss quadrature precision degree) and show that under some specific conditions the two foresaid formulas can be connected to the current weighted quadrature rules. The best application of the case m = 1 in the second formula is when λ is a known root of the function f(x  ). For instance, ∫αβρ(x)(∫λxg(t)dt)dx and ∫αβρ(x)(x-λ)g(x)dx are two samples in which f(λ) = 0. Finally, we present various analytic examples of above rules and introduce a more general form of the mentioned formulas as∫αβρ(x)(f(x)-Pm-1(x;f))dx=∑i=1n∑j=0kdi(m+j)f(m+j)(ri)+Rn(m,k)[f].