bibkey: barry2013riordanpascal authors: Paul Barry year: 2013 title: A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.pdf claim: Conjecture 32 identifies the finite kernel matrix with a Riordan Gram matrix and gives determinant one. strata_touched:

• D5/S1/Recurrence/Algebraic/BarryRiordanPascalKernelDeterminant license: citation-only triage: anchor

Barry’s generalized Pascal matrices

Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences 16 (2013), Article 13.5.4. The journal lists no DOI, and no arXiv version was found.

Verified locator

• URL: https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.pdf
• Section 1, printed page 1, defines the generalized Pascal matrix.
• Section 6, printed page 19, defines the row polynomials and the bivariate coefficient-array convention, then compares the Christoffel-Darboux matrix with the polynomial kernel.
• Conjecture 32 is on printed page 20.

Source statement

Section 1, printed page 1:

In this note, we investigate some properties of the two-parameter generalized Pascal triangle M^{(m)}(a, b) given by the Riordan array M^{(m)}(a, b) = ( 1/(1 − ax) , x(1 + bx)/(1 − ax)^m ).

Section 6, printed page 19:

We can regard the matrix M^(m)(a, b) as the coefficient array of a family of polynomials P_n(x; m, a, b). We have P_n(x; m, a, b) = Σ_{k=0}^{n} T^{(m)}{n,k}(a, b) x^k. For each n, the Christoffel-Darboux quotient ( P{n+1}(x; m, a, b)P_n(y; m, a, b) − P_n(x; m, a, b)P_{n+1}(y; m, a, b) ) / (x − y) is the bi-variate generator of a (n + 1) x (n + 1) matrix D_n(m, a, b).

Later on printed page 19:

It is interesting to compare the matrices defined by the quotient above with those defined by Σ_{j=0}^{n} P_j(x; m, a, b)P_j(y; m, a, b). In the case of monic orthogonal polynomials, such expressions lead to the same matrices. However, M^{(m)}(a, b) is not in general the coefficient array of a family of monic orthogonal polynomials, and hence the two family of matrices will in general differ. In the case m = 2, a = b = 1, we can conjecture the following.

Conjecture 32, printed page 20:

Conjecture 32. The matrix D̃_n(2, 1, 1) with generating function Σ_{k=0}^{n} P_j(x; 2, 1, 1)P_j(y; 2, 1, 1) is given by M_n^{(2)}(a, b)^t M_n^{(2)}(a, b). We then have |D̃_n(2, 1, 1)| = 1 for n ≥ 0. In the above the notation M_n denotes the matrix formed from the first (n + 1) rows and columns of M.

Section 2 defines a Riordan array (g,f) by saying that its i-th column is generated by g(x)f(x)^i, with the first column indexed by zero. Thus the entry T^{(m)}_{n,k}(a,b) is the coefficient of x^n in g(x)f(x)^k.

Reading and formal scope

The page-20 lower summation index is printed as k=0, while the summand uses P_j. The preceding page prints the same kernel with j=0, so the formal sum uses j=0,...,n. The right side of Conjecture 32 retains generic (a,b) in its typography, but the immediately preceding sentence and the left side both fix m=2 and a=b=1; the formal matrix therefore uses (a,b)=(1,1). The generic reading already fails at n=1.

The phrase “with generating function” uses the coefficient-array convention stated immediately above for the Christoffel-Darboux quotient: matrix entry (i,k) is the coefficient of x^i y^k. The formal bivariate polynomial is a polynomial in x whose coefficients are polynomials in y. Finally, T_{n,k} is the coefficient of g f^k, P_n is the row polynomial, and M_n is the first n+1 rows and columns. The result concerns every natural n, including zero, and does not assert Conjecture 31.

The first six rows printed on page 2 are [1], [1,1], [1,4,1], [1,9,7,1], [1,16,26,10,1], and [1,25,70,52,13,1]; they agree with OEIS A114188.

The preregistered literature check found five citing works through OpenAlex. Three arXiv full texts had zero Christoffel hits. The two works without arXiv full text were checked only by title and abstract and remain ASSUMED-UNVERIFIED. OEIS A114188 records the article but no determinant or Gram-matrix settlement; MathDB searches found no entry. A Semantic Scholar request returned HTTP 429 and remains ASSUMED-UNVERIFIED. These bounded searches do not establish exhaustive publication absence or publication priority.