Abstract
Full Text
V. B. OZHEGOV
ON SOME EXTREMAL PROPERTIES OF GENERALIZED APPELL POLYNOMIALS
(Presented by Academician S. N. Bernstein, 12 VI 1964)
1°. A sequence of polynomials \(\{P_n(x)\}_0^\infty\) of degree exactly \(n\) will be called a sequence of generalized Appell polynomials of class \(A^{(k)}\) if
\[ P_n^{(k)}(x)=P_{n-k}(x)\quad (n=k,\ k+1,\ k+2,\ldots). \tag{1} \]
It follows from (1) that if \(\{P_n(x)\}_0^\infty\in A^{(k)}\), then the sequences \(\{P_{n,j}(x)\}_{n=0}^\infty\) \((j=0,\ldots,k-1)\), where
\[ P_{n,0}(x)\equiv P_n(x),\qquad P_{n,j}(x)\equiv P_{n+j}^{(j)}(x) \tag{2} \]
\[ (j=1,\ldots,k-1;\ n=0,1,2,\ldots), \]
also belong to the class \(A^{(k)}\).
We shall also say (which is equivalent to definition (1)) that \(\{P_n(x)\}_0^\infty\in A^{(k)}\) if there exist (formally) \(k\) power series (in what follows—generating functions)
\[ A_m(t)=\sum_{l=0}^{\infty} a_{l,m}t^l\qquad (m=0,\ldots,k-1) \]
such that (also formally)
\[ \sum_{m=0}^{k-1} A_m(t)e^{\varepsilon_m tx}=\sum_{n=0}^{\infty} P_n(x)t^n, \]
where \(\varepsilon_m\) are the primitive \(k\)-th roots of unity.
Theorem 1. In order that \(\{P_n(x)\}_0^\infty\in A^{(k)}\), it is necessary and sufficient that the conditions
\[ \int_0^\infty P_n^{(m)}(x)\,d\gamma(x)=\alpha_{n,m}, \]
hold, where \(\gamma(x)\) is a function of bounded variation on \((0,\infty)\), all moments \(\gamma_n\) of which exist and \(\gamma_0\ne0\); \(\alpha_{n,m}\) \((m=0,\ldots,n;\ n=0,1,2,\ldots)\) are the elements of an infinite triangular table satisfying the condition
\[ \alpha_{n+k,m+k}=\alpha_{n,m}\quad (m=0,\ldots,n;\ n=0,1,2,\ldots). \]
This theorem generalizes results of C. J. Thorne \((^1)\) for sequences of Appell polynomials (class \(A^{(1)}\)).
Corollary 1. The generating functions of a sequence \(\{P_n(x)\}_0^\infty\in A^{(k)}\) have the form
\[ A_m(t)= \frac{ \sum_{n=0}^{\infty}\left[\sum_{l=0}^{k-1}\Delta_{l+1,m+1}^{(k)}\alpha_{n+l,l}\right]t^n }{ \Delta^{(k)}\sum_{n=0}^{\infty}\varepsilon_m^n\frac{\gamma_n}{n!}t^n }, \]
where \(\Delta^{(k)}\) is the Vandermonde determinant \(W(\varepsilon_0,\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_{k-1})\); \(\Delta^{(k)}_{i,j}\) are the algebraic complements of its elements.
For example, for the sequence \(\{S_n(x)\}_0^\infty \in A^{(2)}\) of Euler–Bernstein polynomials ((\(^{2}\), p. 497), the generating functions have the form
\[ A_0(t)=\frac{1+e^{-t}}{e^t+e^{-t}}, \qquad A_1(t)=\frac{e^t-1}{e^t+e^{-t}}, \]
i.e.,
\[ \frac{\operatorname{sh} tx+\operatorname{ch} t(1-x)}{\operatorname{ch} t} = \sum_{n=0}^{\infty} S_n(x)t^n . \]
Corollary 2. Any polynomial \(P_n(x)\) from the sequence
\(\{P_n(x)\}_0^\infty \in A^{(k)}\) can be represented in the form
\[ P_{mk+l}(x)= \sum_{s=0}^{k-1}\sum_{i=r}^{m} \frac{p^{(0)}_{ik+l-s,s}}{(mk-ki+s)!}\, x^{mk-ki+s}, \tag{3} \]
where \(l=0,\ldots,k-1;\ m=0,1,2,\ldots;\ r=0\) when \(l-s \ge 0\); \(r=1\) when \(l-s<0\); \(p^{(0)}_{m,j}\) \((j=0,\ldots,k-1)\) are the constant terms of the polynomials (2).
In what follows we shall assume that, for all \(n\),
\[ P_n^{(n)}(x)=1. \tag{4} \]
\(2^\circ\). Following S. N. Bernstein ((\(^{2}\), p. 515), consider the class \(Ц_{\lambda,1}^{(j)}\) of regularly monotone polynomials on \([0,1]\) with type numbers \(\lambda_1=\lambda-j\), \(\lambda_{2k}=1\), \(\lambda_{2k+1}=\lambda\). From the generalized theorem of S. N. Bernstein ((\(^{2}\), p. 515) it follows that, among all polynomials of the class under consideration satisfying (4), the one that deviates least from zero on \([0,1]\) is the one satisfying the conditions
\[ P_n^{(i)}(0)=0, \qquad i \not\equiv \lambda \pmod{\lambda+1}; \]
\[ P_n^{(i)}(1)=0, \qquad i \equiv \lambda \pmod{\lambda+1}. \tag{5} \]
Consider the sequence \(\{P_n(x)\}_0^\infty\) of extremal polynomials of the class \(Ц_{\lambda,1}^{(0)}\). By condition (5), each polynomial can be represented in the form of an Abel–Goncharov integral
\[ P_n(x)= \int_{\alpha_0}^{x} dx_1 \int_{\alpha_1}^{x_1} dx_2 \int_{\alpha_2}^{x_2} dx_3 \cdots \int_{\alpha_{n-1}}^{x_{n-1}} dx_n, \]
where \(\alpha_i=0\) when \(i \not\equiv \lambda \pmod{\lambda+1}\) and \(\alpha_i=1\) when \(i \equiv \lambda \pmod{\lambda+1}\).
It follows from this representation that the polynomials under consideration satisfy (1) for \(k=\lambda+1\), i.e.,
\(\{P_n(x)\}_0^\infty \in A^{(\lambda+1)}\), and for \(P_n(x)\) the representation (3) is valid with \(k=\lambda+1\). Moreover, the polynomials of the sequences
\(\{P_{n,j}(x)\}_{n=0}^{\infty}\), defined by condition (2), also deviate least from zero among polynomials of the class \(Ц_{\lambda,1}^{(j)}\), and for them one can write a representation analogous to (3). However, representation (3) contains the constant terms of the polynomials of all the sequences \(\{P_{n,j}(x)\}_{n=0}^{\infty}\) \((j=0,\ldots,k-1)\). From the conditions (5), however, it follows that, for the extremal sequences, the only constant terms different from zero will be those of the sequence \(\{P_{n,\lambda}(x)\}_{n=0}^{\infty}\).
Thus, the following is true.
Theorem 2. Among all polynomials of degree \(n\) from the class \(Ц_{\lambda,1}^{(j)}\) of the form
\[ P_n(x)=\pm\frac{x^n}{n!}+p_1x^{n-1}+p_2x^{n-2}+\ldots+p_n \]
deviates least from zero on \([0,1]\) is the polynomial \(\pm P_{n,j}(x)\), where
\[ P_{n,j}(x)=\frac{x^n}{n!}+\sum_{i=1}^{m} \frac{p^{(0)}_{(\lambda+1)i+l+j-\lambda,\lambda}} {[(\lambda+1)(m-i)+\lambda-l]!}\, x^{(\lambda+1)(m-i)+\lambda-j} \]
\[ (j=0,\lambda,\ldots;\quad l=-j,\ldots,-j+\lambda;\quad n=m+(\lambda+1)+l;\quad m=1,2,3,\ldots), \]
whose coefficients are determined from the conditions
\[ \frac{1}{n!}+\sum_{i=1}^{m} \frac{p^{(0)}_{(\lambda+1)i+l,\lambda}} {[(\lambda+1)(m-i)]!}=0 \]
\[ (n=m(\lambda+1)+l;\quad l=-\lambda,\ldots,0;\quad m=1,2,3,\ldots), \]
and this least deviation is equal to
\[ L_n^{(\lambda)}=\bigl|p^{(0)}_{n,\lambda}\bigr|,\qquad L_n^{(j)}=\left| \frac{1}{n!}+\sum_{i=1}^{m} \frac{p^{(0)}_{(\lambda+1)i+l+j-\lambda,\lambda}} {[(\lambda+1)(m-i)+\lambda-j]!} \right| \]
\[ (j=0,\ldots,\lambda-1). \]
Corollary. The generating functions of the sequence \(\{P_n(x)\}_0^\infty\) of extremal polynomials of the class \(\mathcal{Ц}^{(0)}_{\lambda,1}\) have the form:
\[ A_0(t)=1+\frac{1-e^t}{\sum_{j=0}^{\lambda}\varepsilon_j t}, \qquad A_m(t)=\varepsilon_m\frac{1-e^t}{\sum_{j=0}^{\lambda}\varepsilon_j t} \quad (m=1,\ldots,\lambda). \]
For example, for the extremal sequence \(\{P_n(x)\}_0^\infty\in \mathcal{Ц}^{(0)}_{2,1}\) \((({}^{2}),\ p. 515)\), the generating functions have the form
\[ A_0(t)=1+\psi(t),\qquad A_1(t)=\varepsilon_1\psi(t),\qquad A_2(t)=\varepsilon_2\psi(t), \]
where
\[ \psi(t)=\frac{1-e^t}{e^t+2e^{-1/2t}\cos\frac{\sqrt{3}}{2}t}. \]
Consequently,
\[ e^{tx}+\bigl(e^{tx}+\varepsilon_1 e^{\varepsilon_1tx}+\varepsilon_2 e^{\varepsilon_2tx}\bigr)\psi(t) =\sum_{n=0}^{\infty}P_n(x)t^n, \]
whence
\[ 1+3\psi(t)=\sum_{n=0}^{\infty}p^{(0)}_{n,2}t^n. \]
Thus, the coefficients in the expansion of the function \(1+3\psi(t)\) in powers of \(t\), up to sign, coincide with the quantities \(L_n^{(2)}\) of the deviations of polynomials of degree \(n\) of the class \(\mathcal{Ц}^{(2)}_{2,1}\). Hence, in particular, there follows S. N. Bernstein’s formula \((({}^{2}),\ p. 516)\)
\[ \sum_{n=0}^{\infty}p^{(0)}_{3n,2}t^{3n} = \frac{3}{e^t+2e^{-1/2t}\cos\frac{\sqrt{3}}{2}t}. \]
Leningrad Mechanical Institute
Received
1 VI 1964
REFERENCES
- C. J. Thorne, Am. Math. Monthly, 52, 191 (1945).
- S. N. Bernstein, Collected Works, 2, No. 100, Publishing House of the Academy of Sciences of the USSR, 1954.