Inverses of Motzkin and Schröder paths (Q2855548)

In this paper the author investigates the connection between the generating function (and its inverse) of the number of Motzkin paths with weighted horizontal steps and Chebyshev polynomials of the second kind. The author also considers Motzkin numbers of paths in a band, i.e. when the height of the path is limited (and by definition staying above the \(x\)-axis).NEWLINENEWLINENEWLINE It starts with introducing the notion of Motzkin path and weighted steps. The weighted Motzkin numbers can be seen to satisfy a recurrence relation. The next step in the paper is to reveal the relation between the compositional inverse of the generating function and Chebyshev polynomials of the second kind. The paper then continues investigating Hankel determinants.NEWLINENEWLINENEWLINE Next, Motzkin paths are generalised to paths with horizontal step length \(w\). Specialising to \(w\) equal to 2 one gets Schröder paths and the author obtains some results for these numbers in terms of Chebyshev polynomials. All in all the paper reveals some interesting results on generating functions and explicit formulas for Motzkin numbers though not all stated results are correct and when using them the reader is advised to check toroughly. Below I state some flaws.NEWLINENEWLINENEWLINE The top formula on page 4 involving the Catalan numbers is wrong, the factor \((t+1)\) in this formula should be replaced by \(1/(t+1)\). I verified the formulas on page 5 against results to be found in the standard work on mathematical functions by \textit{M. Abramowitz} and \textit{I. A. Stegun} [Handbook of mathematical functions. New York etc.: John Wiley \& Sons (1972; Zbl 0543.33001)] and everything checked out to be okay, except for a missing \(x\) in the recurrence relation for the Gegenbauer polynomials, the first term should be \(C^{\lambda}_{n-1}(x) \) instead of \(C^{\lambda}_{n-1} \).NEWLINENEWLINENEWLINE On page 6 the formula for \(\sum_{j \geq 0} x^j \sum_{i \geq j} m_{ij}t^i \) is wrong, the line should be replaced by NEWLINE\[NEWLINE \sum_{j \geq 0} x^jt^j(\phi(t)^{-(j+1)}) = \frac{1}{\phi(t)-xt}= U(t,-\frac{\beta-x}{2}).NEWLINE\]NEWLINE Also the formula for \(\sum t^{-j}\sum_{i \geq j}m_{ij}x^i \) is lacking a minus sign and the final result should read \(U(x,-\frac{\beta-1/t}{2})\).NEWLINENEWLINENEWLINE The formula on the bottom of page 7 is wrong, the first factor should read \( |M_{i+j;\beta}| \) instead of \( |M_{i+j;1}| \).NEWLINENEWLINENEWLINE Also I think the result for the Hankel determinants on page 8 is incorrect, the powers \(n+1\) should be \(-(n+1)\) and the second term should be subtracted instead of added. The formula obtained by taking \( a = \sin^2\theta \) and \( b=\cos\theta-1\), I could again verify against Abramowitz and Stegun [loc. cit, Formula 22.3.16].NEWLINENEWLINENEWLINE The determinants on page 9 contain the same flaws, in the formula under 3, the \(-1\) should be replaced by \(-1\) to the power \(n+1\), and in 4, the \(+\) sign for the second term should be a \(-\) sign. The results obtained by taking \(a=b=1=\beta\) in 3 and \(a=0\), \(b=1\), \(\beta=2\) in 4 can be obtained more easily by observing that these values lead to a generating function being equal to \(\frac{1}{1-2t+t^2}\), according to the generating function given on page 8. Then it is easily seen that the coefficent of \(t^n\) is \(n+1\).NEWLINENEWLINENEWLINE In the last function on page 9 it is unclear what the author means by \(M_n^{(k)}\). Probably the author means \(M_{n,\beta}^{(k)}\), otherwise the LHS of the formula is independent of \(\beta\), whereas the RHS is dependent on \(\beta\) via the \(\mu\)-function. In the first table of page 10 it is unclear what \(M_n^{(4)}\) is, it clearly cannot be \(M_{n,1}^{(4)}\) because otherwise the last two entries in row zero are contradicting the last two entries listed in 4 at the bottom of page 10. In Lemma 4 the term \(-2xt^2\) seems to be wrong. According to me it should be \(-2xt\).NEWLINENEWLINENEWLINE The first table on page 15 contains an error. The \(n=3\), \(j=1\) entry should be \(2+2\beta\) instead of \(2+\beta\). Moreover, I believe the generating function for the compressed Schröder number is wrong, it should read \((\frac{(\hat{z}-\sqrt{\hat{z}^2-1})^{j+1}}{\sqrt t} \), resulting in a formula for the double sum being \(\frac{1-\beta t - 2xt - \sqrt{(1-\beta t)^2-4t}}{2t-2x\sqrt t + 2x \beta t\sqrt t + 2tx^2} \). On page 17, the first formula for \(\hat{s}_{n,n-k}\) is lacking a \((-1)^k\). I did not check the second equality in this expression. I did not check the examples following Theorem 5.