Multiple orthogonal polynomials associated with confluent hypergeometric functions
Given a pair of measures \( (\mu_{0}, \mu_{1})\) supported on the real line, type I multiple orthogonal polynomials for \(\vec{n}=(n_{0}, n_{1})\) are given by a vector \(( A^{(0)}_{\vec{n}}(x), A^{(1)}_{\vec{n}}(x))\) of polynomials with deg \(A^{(j)}_{\vec{n}}(x) \leq n_{j} -1, j=0,1\) satisfying the orthogonality and normalisation conditions \(\int x^j A^{(0)}_{\vec{n}}(x) d\mu_{0} + \int x^j A^{(1)}_{\vec{n}}(x) d\mu_{1}=0,\) if \(0\leq k\leq |\vec{n}| -2,\) and \(=1,\) if \(k= |\vec{n}| -1 . \) On the other hand, type II multiple orthogonal polynomials for \(\vec{n}=(n_{0}, n_{1})\) are monic polynomials \(P_{\vec{n}}(x)\) of degree \(|\vec{n}|\) which satisfy the orthogonality conditions \(\int x^k P_{\vec{n}}(x) d\mu_{j}=0, j=0, 1, k= 0,1, \cdots, n_{j}-1.\) The above orthogonality conditions give a non-homogeneous system of \(|\vec{n}|\) linear equations for the unknown coefficients of the polynomials \( A^{(0)}_{\vec{n}}(x), A^{(1)}_{\vec{n}}(x)\) and \(P_{\vec{n}}(x),\) respectively. For more information about multiple orthogonal polynomials, see [\textit{E. M. Nikishin} and \textit{V. N. Sorokin}, Rational approximations and orthogonality. Transl. from the Russ. by Ralph P. Boas. Transl. ed. by S. Gelfand. Providence, RI: American Mathematical Society (1991; Zbl 0733.41001)], as well as Chapter 23 in [\textit{M. E. H. Ismail}, Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. Cambridge: Cambridge University Press (2005; Zbl 1082.42016)]. In the paper under review the authors study multiple orthogonal polynomials with respect to two absolutely continuous measures supported on the positive real line and having an integral representation in terms of weight functions \(\mathcal{W}(x;a,b;c)\) and \(\mathcal{W}(x;a, b;c+1),\) where \(\mathcal{W}(x;a,b;c)= \frac{\Gamma(c)}{\Gamma(a) \Gamma(b)} e^{-x} x^{a-1} \textbf{U }(c-b, a-b+1;x)\) with \(a,b,c\) positive real numbers, \(c> max(a, b)\) and \(\textbf{U}( \alpha, \beta; x)\) is the Tricomi function, also called confluent hypergeometric function of the second kind.\\ As a first result, it is proved that the above weight functions constitute a Nikishin system (see [\textit{E. M. Nikishin}, Math. USSR, Sb. 41, 409--425 (1982; Zbl 0478.30007)]) As a consequence, the multiple orthogonal polynomials of both type I and type II with respect to those weights exist and are unique for every multi-index (i. e. they constitute a perfect system). A Rodrigues-type formula for type I polynomials is obtained. Moreover, differential properties for type I and type II multiple orthogonal polynomials on the step line are deduced. On the other hand, several characterisations of type II orthogonal polynomials on the step-line (i. e. the choice of multi-indices \((n,n), (n+1)\) in order to define the sequence of \(2\)-orthogonal polynomials \(P_{2n}(x)= P_{n,n} (x),\) \(P_{2n+1}(x)= P_{n+1, n}(x),\) respectively, see [\textit{P. Maroni}, Ann. Fac. Sci. Toulouse, Math. (5) 10, No. 1, 105--139 (1989; Zbl 0707.42019)] are presented. First, they satisfy the Hahn property, i. e. the sequence of their first derivatives is also a \(2\)-orthogonal sequence. Furthermore, the authors prove that they have a representation as a generalized \(_{2}F_{2}\) hypergeometric series, they satisfy a third order linear differential equation as well as a four term recurrence relation. Each of the sequences of recurrence coefficients is unbounded and asymptotically periodic of period \(2.\) On the other hand, they find limiting relations between these polynomials and multiple orthogonal polynomials with respect to weight functions involving the modified Bessel function of second kind (see [\textit{W. Van Assche} and \textit{S. B. Yakubovich}, Integral Transforms Spec. Funct. 9, No. 3, 229--244 (2000; Zbl 0959.42016)]. Finally, \(3\)-fold symmetric (also called \(2\)-symmetric) polynomial sequences are analyzed. This means that given a polynomial sequence \(\{P_{n}(x)\}_{n\geq0}\) there exist three polynomial sequences \(\{P^{[k]}_{n}(x)\}_{n\geq0}, k=0, 1, 2,\) (the so called cubic components of the sequence \(\{P_{n}(x)\}_{n\geq0}\)) such that \(P_{3n+k}(x)= x^k P^{[k]}_{n}(x^3), k= 0,1, 2.\) Then it is proved that all three cubic components of \(3\)-fold symmetric Hahn-classical \(2\)-orthogonal polynomials are themselves Hahn-classical. The motivation of the results as well as the presentation of this paper are very remarkable.
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