Transactions of A. Razmadze Mathematical Institute Vol. 175 (2021), issue 2, 235–251 RATIONAL P ´ AL TYPE (0, 1; 0)-INTERPOLATION AND QUADRATURE FORMULA WITH CHEBYSHEV–MARKOV FRACTIONS SHRAWAN KUMAR 1 , NEHA MATHUR 2 , VISHNU NARAYAN MISHRA 3∗ AND PANKAJ MATHUR 1∗ Abstract. We present a P´al-type (0, 1; 0)-interpolation on an inter-scaled set of nodes, when Her- mite and Lagrange data are prescribed on the zeros of Chebyshev–Markov sine fraction Un(x) and its derivative U ′ n (x), respectively. A quadrature formula based on the obtained P´al-type interpolation has been constructed. Coefficients of this quadrature are obtained in the explicit form. 1. Introduction The study of different type interpolation processes has been a subject of interest for several math- ematicians. In almost all the cases the interpolatory polynomials are considered on the nodes which are the zeros of certain classical orthogonal polynomials. The main idea of the present paper is to construct a rational interpolation process and its corresponding quadrature formula. Let R 2n−1 (a 0 ,a 1 ,a 2 ,...,a 2n−1 ) be a rational space defined as R 2n−1 (a 0 ,a 1 ,...,a 2n−1 ) :=  p 2n−1 (x)  2n−1 k=0 (1 + a k x)  , where p 2n−1 (x) is a polynomial of degree ≤ 2n − 1 and {a k } 2n−1 k=0 are real and belong to [−1, 1], or are paired by a complex conjugation. Chebyshev and Markov introduced rational cosine and sine fractions [9] which generalize Chebyshev polynomials, possess many similar properties [8, 16, 18] and are called Chebyshev–Markov rational fractions. More details on the rational generalization of Chebyshev polynomials can be found in [1–6, 19]. Let U n (x) be the rational Chebyshev–Markov sine fraction, U n (x)= sin µ 2n (x) √ 1 − x 2 , (1.1) where µ 2n (x)= 1 2 2n−1  k=0 arccos x + a k 1+ a k x , µ ′ 2n (x)= − λ 2n (x) √ 1 − x 2 , λ 2n (x)= 1 2 2n−1  k=0  1 − a 2 k 1+ a k x , n ∈ N. (1.2) The rational fraction U n (x) can be expressed as U n (x)= P n−1 (x)  Π 2n−1 k=0 (1 + a k x) , where P n−1 (x) is an algebraic polynomial of degree n − 1 with a real coefficient, and {a k } 2n−1 k=0 are as defined above. The fraction U n (x) has n − 1 zeros on the interval (−1, 1) given by −1 <x n−1 <x n−2 < ··· <x 2 <x 1 < 1, 2020 Mathematics Subject Classification. Primary 05C38, 15A15, Secondary 05A15, 15A18. Key words and phrases. P´al-typeinterpolation; Lobatto-type quadrature; Rational space; Chebyshev–Markov fractions. ∗ Corresponding author.