On numerical integration methods with T-distribution weight function

Abstract A class of weighted quadrature rules whose weight function corresponds to T distribution, i.e. K ( 1 + x 2 ) - ( p - 1 2 ) , x ∈ ( - ∞ , ∞ ) , is introduced and investigated. The integration formulas, given in this work, are generally in the following form: ∫ - ∞ ∞ ( 1 + x 2 ) - ( p - 1 2 ) f ( x ) d x = ∑ i = 1 n w i f ( x i ) + R n [ f ] , where x i is the zeros of orthogonal polynomials with respect to the introduced weight function, w i is the related coefficient and R n [ f ] is the error function. It is important to point out that the above mentioned formula is valid only for the finite values of n . In other words, p  > {max  n } + 1 must be satisfied in order that the above integration formula is applicable. Some analytical examples are finally given and compared.