A Note on Finite Quadrature Rules with a Kind of Freud Weight Function

We introduce a finite class of weighted quadrature rules with the weight function | x | − 2 𠑎 e x p ( − 1 / 𠑥 2 ) on ( − ∞ , ∞ ) as ∫ ∞ − ∞ | 𠑥 | − 2 𠑎 e x p ( − 1 / 𠑥 2 ∑ ) 𠑓 ( 𠑥 ) 𠑑 𠑥 = 𠑛 𠑖 = 1 𠑤 𠑖 𠑓 ( 𠑥 𠑖 ) + 𠑅 𠑛 [ 𠑓 ] , where 𠑥 𠑖 are the zeros of polynomials orthogonal with respect to the introduced weight function, 𠑤 𠑖 are the corresponding coefficients, and 𠑅 𠑛 [ 𠑓 ] is the error value. We show that the above formula is valid only for the finite values of 𠑛 . In other words, the condition 𠑎 ≥ { m a x 𠑛 } + 1 / 2 must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared.