Support Math 4 Wisdom Study Groups Featured Investigations Featured Projects Contact Andrius Kulikauskas m a t h 4 w i s d o m @ g m a i l . c o m +370 607 27 665 Eičiūnų km, Alytaus raj, Lithuania Thank you, Participants! Thank you, Veterans! Dave Gray Francis Atta Howard Jinan KB Christer Nylander Kirby Urner Thank you, Commoners! Free software Open access content Expert social networks Patreon supporters Jere Northrop Daniel Friedman John Harland Bill Pahl Anonymous supporters! Support through Patreon! Kim-Zeng Trees Combinatorial Interpretation of Orthogonal Sheffer Polynomial Coefficients I am interpreting the combinatorial objects described in Dongsu Kim, Jiang Zeng. A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials. Kim and Zeng write the recurrence relation as follows Kim and Zeng: {$P_{n+1}(x)=[x-(\alpha\beta + nu_3+nu_4)]P_n(x)- u_1u_2 n(n-1 +\beta )P_{n-1}(x)$} Kim and Zeng: {$P_{n+1}(x)=[x - (u_3+u_4)n + \alpha\beta)]P_n(x)-[u_1u_2n(n-1) + u_1u_2\beta n]P_{n-1}(x)$} And here it is in my notation Andrius: {$P_{n+1}(x)=[x-(ln+cf)]P_n(x)-[kn(n-1) + nc]P_{n-1}(x)$} Andrius: {$P_{n+1}(x)=[x-(((-\alpha)+(-\beta))n+cf)]P_n(x)-(-\alpha)(-\beta)[n(n-1) + nc]P_{n-1}(x)$}
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