caa a22 4500 605461619 CHVBK 20210128100244.0 cr unu---uuuuu 210128e20150301xx s 000 0 eng 10.1007/s10114-015-4209-5 doi (NATIONALLICENCE)springer-10.1007/s10114-015-4209-5 Context-free grammars for triangular arrays [Elektronische Daten] [Robert Hao, Larry Wang, Harold Yang] We consider context-free grammars of the form $G = \{ f \to f^{b_1 + b_2 + 1} g^{a_1 + a_2 } ,g \to f^{b_1 } g^{a_1 + 1} \} $ , where a i and b i are integers subject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T(n, k)}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let T n (x) = Σ k=0 n T(n, k)x k . Based on the differential operator with respect to G, we define a sequence of linear operators P n such that T n+1(x) = P n (T n (x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Brändén, we obtain a necessary and sufficient condition for the operator P n to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh. Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg, 2015 Context-free grammar nationallicence stable polynomials nationallicence real-rootedness nationallicence Bessel numbers nationallicence Whitney numbers nationallicence Hao Robert Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P. R. China aut Wang Larry Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P. R. China aut Yang Harold Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P. R. China aut Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/3(2015-03-01), 445-455 1439-8516 31:3<445 2015 31 10114 https://doi.org/10.1007/s10114-015-4209-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10114-015-4209-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Hao Robert Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P. R. China aut NATIONALLICENCE 700 1- Wang Larry Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P. R. China aut NATIONALLICENCE 700 1- Yang Harold Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P. R. China aut NATIONALLICENCE 773 0- Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/3(2015-03-01), 445-455 1439-8516 31:3<445 2015 31 10114