(1) Duality Triads
Let us consider the infinite array of numbers cn,k satisfying the following recurrence equation:
cn+1,k = ik-1 cn,k-1 + qk cn,k + dk+1 cn,k+1
where c0,0 = 1, c0,k = 0 for k>0.
Let now {Φn}n≥0 be a polynomial sequence (i.e. deg(Φn(x)) = n) determined by the following recurrence equation "dual with respect to do (1)":
xΦk(x) = dk Φk-1(x) + qk Φk(x) + ik Φk+1(x)
where Φ0(x) = 1, Φ-1(x) = 0 for k ≥ 0.
then as it was shown in [akk1]:
(2) References
-
[akk1] A. K. Kwaśniewski, On duality triads, Bull. Soc. Sci. Letters Łódź 53, Rech. sur def. 42 (2003), 11.
-
[akk2] A. K. Kwaśniewski, On Fibonomial and other triangles versus duality, Bull. Soc. Sci. Letters Łódź 53, Rech. sur def. 42 (2003), 27.
-
[sch1] M. Schork, On a generalization of duality triads,
CEJM 4(2) 2006 304-318
-
[sch1] M. Schork, Duality triads of higher rank: Further properties and some examples, CEJM 4(3) 2006 507-524
