Recurrence relations for polynomial sequences via Riordan matrices

• The paper gives a general recurrence for polynomial sequences naturally associated with Riordan matrices and characterizes Riordan arrays by their rows.The main recurrence follows from the construction algorithm.Source: §1 · pp. 2–4
• The Pascal triangle motivates interpreting Riordan matrices by rows, with its combinatorial recurrence connecting row and column interpretations and inspiring a generalization of the recurrence rule.The paper presents Algorithm 1 as this generalization.Source: §1 · p. 4
• For Fibonacci, Pell, and Morgan–Voyce polynomials, associated Riordan matrices determine the sequences, and matrix products recover known relationships between families.The examples illustrate how Riordan-matrix structure appears in the families’ recurrence relations.Source: §1 · p. 4
• The framework extends the examples, recovers polynomial generating functions through Riordan-array action, and represents usual umbral composition as a Riordan-matrix product.Source: §1 · p. 4
• Hadamard products extend the recurrence framework to generalized Appell polynomial sequences in generalized umbral calculus.These recurrences follow from the main recurrence and construction algorithm.Source: §1 · p. 4

• Riordan arrays provide a way to pass from a column description to a row description.Source: §2 · p. 5
• The Pascal polynomials satisfy the recurrence p_n+1(x) = (x + 1)p_n(x), illustrating a row-based description.Source: §2 · pp. 5–6
• The Fibonacci polynomials are associated with T(1|1 − x^2), whose triangle follows d_n,k = d_n−2,k + d_n−1,k−1 for k > 0.Source: §2 · p. 6
• The Pell polynomials are related to Fibonacci polynomials by P_n(x) = F_n(2x).Source: §2 · pp. 7–8
• The two Morgan–Voyce triangles share a construction rule but have different initial conditions, and Riordan-array products interpret relationships between their polynomial sequences.Source: §2 · pp. 8–9

3.1. The main theorem.

• A lower-triangular matrix is a Riordan matrix, or an arithmetical triangle in the sense of, exactly when its associated polynomials satisfy the stated recurrence for sequences with g_0 ≠ 0.In that case the matrix is T(f|g); the associated polynomial coefficients are taken from its rows, and the sequence is an ordinary polynomial sequence when the diagonal entries are nonzero.Source: §3.1 · p. 10
• When g(x) has finite degree m, the recurrence from Theorem 5 is eventually finite.The passage states this for a polynomial g(x) whose leading coefficient g_m is nonzero.Source: §3.1 · p. 12
• Under umbral composition, multiplication of Riordan matrices corresponds to composition of their associated polynomial sequences; the resulting operation on the specified sequence set forms a group isomorphic to the Riordan group.Source: §3.1 · p. 14

3.2. Some relationships between polynomials sequences of Riordan type and some classical examples.

• Left multiplication by T(h | 1) corresponds to fixed elementary row operations on the matrix T(f | g), determined by h’s coefficients; for h(x)=a+bx, q0(x)=ap0(x) and qn(x)=bpn−1(x)+apn(x).Source: §3.2 · p. 16
• A small perturbation of the first-kind Chebyshev sequence is represented by a Riordan matrix, and the matrix operations recover a relation between Tn(x) and Un(x).The generating function obtained is identified as that of the classical first-kind Chebyshev polynomials.Source: §3.2 · pp. 17–19
• Row-and-column shifts of Riordan matrices yield recurrences between their associated polynomial families, including families associated to T(fg^n|g).Once the family associated to T(f|g) is known, the paper says those for T(fg^n|g) can be calculated for n in Z.Source: §3.2 · p. 20
• The examples use Riordan arrays to express relationships between the two kinds of Morgan–Voyce polynomials and between Pell and Fibonacci polynomials.The Morgan–Voyce families are identified with arrays T(1|(1 − x)^2) and T(1 − x|(1 − x)^2), respectively; the Pell–Fibonacci relation is also presented through their arrays.Source: §3.2 · pp. 21, 23
• A product of Riordan matrices with affine second components gives a general relation between the associated polynomial families of the product and its factors.Proposition 18 frames the relation through T(l|m) = T(γ|α + βx)T(f|g)T(c|a + bx).Source: §3.2 · p. 22
• The Fermat polynomials are constructed from a Riordan array, and the section relates them to second-kind Chebyshev polynomials and relates Boubaker polynomials to both families.The Boubaker discussion identifies a relation to second-kind Chebyshev polynomials and then to Fermat polynomials.Source: §3.2 · pp. 23–25

Sheffer sequences

• The Sheffer example specializes the preceding results to give recurrence relations for Sheffer polynomials, their coefficients, and their derivatives.A Sheffer sequence is identified through a Riordan matrix T(f | g).Source: §4 · pp. 26, 32, 33

Brenke polynomials

• Applying a fixed T(f | 1) to different series h produces classical polynomial families, including the Brenke family and its stated Appell and reversed Taylor special cases.The section also uses Riordan-matrix relationships to connect classical Sheffer sequences.Source: §4 · p. 27

Pidduck and Mittag-Leffler polynomials

• The examples identify factorially scaled versions of the Pidduck and Mittag-Leffler families and state that the two families are related.The Pidduck example gives explicit polynomials through degree four, while the Mittag-Leffler example displays terms through degree four.Source: §4 · pp. 28–29

Theorem 26

• Theorem 26 characterizes generalized Appell polynomials [2] by an if-and-only-if condition using coefficient sequences f, g, and h and the Hadamard h-weighted associated polynomials of T(f | g).The conditions include f_0, g_0 ≠ 0 and h_n ≠ 0; the theorem also gives recurrences for the family’s coefficients.Source: §4 · p. 30

Corollary 28

• For a Riordan-type polynomial sequence, Corollary 28 relates its Hadamard h-weighted and h′-weighted sequences when h is Hadamard invertible.The corollary expresses the relation using derivatives of the series involved.Source: §4 · pp. 31–32

Laguerre and Hermite examples

• The Laguerre and Hermite examples yield recurrence relations for the polynomials and their coefficients; the Hermite construction also recovers the usual polynomials, their general term, and their parity relation.The Laguerre example uses the Pascal triangle representation; the Hermite polynomials satisfy eHn(−x) = (−1)^n eHn(x).Source: §4 · pp. 34–36

Hadamard-weighted sequences

• For a Hadamard-unit h, the associated map is bijective and transports umbral composition into an operation making Rh a group, with Hh a group isomorphism.This describes how Riordan-group operations transfer to Hadamard h-weighted sequences.Source: §4 · p. 37

Convolution sequences

• A polynomial sequence is a convolution sequence exactly when it is an ex-Hadamard weighted sequence generated by a Riordan sequence associated with T(g | g).The associated Riordan sequence is linked to the Riordan array T(g | g).Source: §4 · pp. 38–39

Convolution sequences and delta operators

• For a convolution sequence associated with T(g | g), the binomial-type sequence rn(x) = n!sn(x) has delta operator x A(x)(D), where A is that array’s A-sequence.Conversely, the sequence derived from the delta operator x A(x)(D) is the convolution sequence associated with T(A | A).Source: §4 · p. 39

Generalized Appell characterization

• A matrix has an associated generalized Appell sequence exactly when its induced linear action on K[x] is given by power series f, g, and h with f0 and g0 nonzero and every hn nonzero.This characterizes generalized Appell sequences through linear transformations in K[x].Source: §4 · p. 40