Mathematics

V. P. KONOPLEV

ON POLYNOMIALS ORTHOGONAL WITH WEIGHT FUNCTIONS VANISHING OR BECOMING INFINITE AT INDIVIDUAL POINTS OF THE INTERVAL OF ORTHOGONALITY

(Presented by Academician I. G. Petrovskii on 6 VII 1961)

Among questions in the application of the theory of orthogonal polynomials, an important place is occupied by the problem of convergence of Fourier–Chebyshev series in polynomials orthogonal with arbitrary weight functions. For a general solution of this problem it is necessary to know the asymptotic properties of the polynomials as their indices increase without bound, and the asymptotics must be valid on the entire interval of orthogonality.

The cases of systems (apart from the classical ones) for which such asymptotics have been obtained are few in number \((^{1,2})\), and in recent years, as noted by Szegő \((^3)\), only slight progress has been made in this direction. Among the known analytic devices for obtaining asymptotic representations there are none suitable for systems of polynomials orthogonal with weight functions that vanish or become infinite at individual points inside the interval of orthogonality.

The apparatus of differential-return equations that we propose is promising in this respect.

Theorem. Let the function \(\rho(x)\) on the interval \([-1,+1]\) be nonnegative and

1) \(\rho(x)=O\left(|x-x_k|^{\gamma_k}\right)\) as \(x\to x_k\), where
\[ -1=x_1<x_2<\cdots<x_r=+1,\quad \gamma_k>-1,\quad k=1,\ldots,r \]
\((\gamma_1=\beta,\ \gamma_r=\alpha)\);

2) \(\rho(x)\ge \rho_0>0\) for all \(x_k+\delta\le x\le x_{k+1}-\delta,\ k=1,\ldots,r-1,\ \delta>0\);

3) the functions \(\rho(x)a(x)\) and \([\rho(x)a(x)]'-\rho(x)b(x)\) are absolutely continuous and vanish at the endpoints of the interval;

\[ \{[\rho(x)a(x)]'-\rho(x)b(x)\}'=P(x)\rho(x), \]

where \(P(x)\) is a polynomial of some degree, and
\[ a(x)=(x^2-1)\prod_{k=2}^{r-1}(x-x_k)^2, \]
\[ b(x)=[(\alpha+\beta+2)x+\alpha-\beta]\prod_{k=2}^{r-1}(x-x_k)^2+ \]
\[ +(x^2-1)\sum_{k=2}^{r-1}\gamma_k(x-x_k)\prod_{\substack{i=2\\ i\ne k}}^{r-1}(x-x_i)^2. \]

If
\[ \widetilde y_n(x)=x^n-S_nx^{n-1}+d_nx^{n-2}+\cdots,\qquad n=0,1,2,\ldots, \]
are polynomials orthogonal with the weight function \(\rho(x)\) on the interval \([-1,+1]\),

then in general they satisfy the differential–recurrence equation

\[ a(x)y''+b(x)y'+c(x,n)y=\sum_{k=1}^{m} A_k \widetilde y_{n-k}(x), \tag{1} \]

where

\[ c(x,n)=-n\left(n+1+\alpha+\beta+\sum_{k=2}^{r-1}\gamma_k\right)x^{2r-4} +q_{2r-5}x^{2r-5}+\cdots+q_0, \]

\[ m=\max[\operatorname{deg} P(x),\,2r-4]. \]

Proof. Denote \(L[y]=a(x)y''+b(x)y'\). Clearly,

\[ L[\widetilde y_n]/\widetilde y_n(x) =-c(x,n)+R_{n-1}(x)/\widetilde y_n(x), \]

where \(R_{n-1}(x)\) is a polynomial of degree not exceeding \(n-1\); therefore

\[ R_{n-1}(x)=\sum_{k=1}^{n} A_k \widetilde y_{n-k}(x). \]

Thus the identity

\[ L[\widetilde y_n]+c(x,n)\widetilde y_n(x) =\sum_{k=1}^{n} A_k \widetilde y_{n-k}(x) \]

holds. Multiply this identity by \(x^l\rho(x)\), \(l=0,\ldots,n-1\), and integrate from \(-1\) to \(+1\). Integrating by parts and taking into account the assumptions of the theorem, together with the equalities

\[ \int_{-1}^{+1}\rho(x)x^{n+j}\widetilde y_n(x)\,dx = B_j\int_{-1}^{+1}\rho(x)\widetilde y_n^{\,2}(x)\,dx, \]

we successively find all \(A_k\), \(k=1,\ldots,n\); moreover it turns out that \(A_k=0\), \(k=m+1,\ldots,n\). The theorem is completely proved.

The conditions of the theorem are satisfied, for example, for the weight functions:

1) \(\rho(x)=a^{\pi(x)}\), \(a>0\), \(\pi(x)\) an arbitrary polynomial;

2) \[ \rho(x)=(1-x)^\alpha(1+x)^\beta\prod_{k=2}^{r-1}|x-x_k|^{\gamma_k}, \quad -1<x_k<+1,\quad \alpha,\beta,\gamma_k>-1,\quad k=2,\ldots,r-1. \]

Writing out the differential–recurrence equation in each case causes no difficulty. Thus, for the weight function

\[ \rho(x)=(1-x)^\alpha(1+x)^\beta|x-x_0|^\gamma, \quad -1<x_0<+1,\quad \alpha,\beta,\gamma>-1, \]

one obtains the equation

\[ a(x)y''+b(x)y'+c(x,n)y=A(n)\widetilde y_{n-1}(x), \tag{2} \]

where

\[ a(x)=(x^2-1)(x-x_0)^2, \]

\[ b(x)=\{(x^2-1)\gamma+[(\alpha+\beta+2)x+\alpha-\beta](x-x_0)\}(x-x_0), \]

\[ c(x,n)=-n(n+1+\alpha+\beta+\gamma)(x-x_0)^2+Q_n(x-x_0)+D_n, \]

\[ A(n)=\frac{\alpha_n}{\alpha_{n-1}} \left[(S_{n+1}-S_n)(2n+\alpha+\beta+\gamma)+2S_{n+1}+\alpha-\beta+\gamma x_0\right], \]

\[ \alpha_k=\int_{-1}^{+1}(1-x)^\alpha(1+x)^\beta|x-x_0|^\gamma y_k^2(x)\,dx. \]

If the square bracket on the right-hand side of the equation vanishes for all \(n=0,1,2,\ldots\), then the equation becomes homogeneous, but this proves to be equivalent to the condition \(Q_n=0\), which in turn leads to the expressions

\[ S_n=-n\,\frac{\alpha-\beta+\gamma x_0}{2n+\alpha+\beta+\gamma},\qquad \widetilde y_n(x_0)\widetilde y_n'(x_0)=0,\qquad \gamma\ne 0. \]

Considering the equation in the cases \(n=1,2\), we see that these conditions are possible only when \(x_0=0\) and \(\alpha=\beta\), i.e., for the case of the weight function

\[ \rho(x)=(1-x^2)^\alpha |x|^\gamma, \]

when the equation has the form

\[ (x^2-1)x^2y''+\{(x^2-1)\gamma+2(\alpha+1)x^2\}xy' \]

\[ -\left[n(n+1+2\alpha+\gamma)x^2+\frac{1-(-1)^n}{2}\gamma\right]y=0. \]

If \(\gamma=0\), then equation (2) becomes the well-known equation for the Jacobi polynomials.

We shall use equation (2) to obtain an asymptotic representation of polynomials orthogonal with the weight function

\[ \rho(x)=(1-x)^\alpha(1+x)^\beta |x-x_0|^\gamma, \]

\[ -1\le x\le +1,\qquad -1<x_0<+1,\qquad \alpha,\beta>-1,\qquad \gamma\ge 0, \]

on the whole interval of orthogonality, according to the scheme developed in our paper (4).

To apply this scheme, we divide the interval \([-1,+1]\) into the parts
\([-1,l_1]\), \([l_1,x_0]\), \([x_0,l_2]\), \([l_2,+1]\), each of which contains one singular point of equation (2) at one of its endpoints. In all cases, by means of a suitable change of the independent variable, we shall write equation (2) in one of the following two forms:

\[ t^2y''+tp(t)y'+[\lambda^2t^2r(t)-q(t)]y=R(t), \]

where \(t=\pm(x-x_0)\), \(t\ge 0\), or

\[ ty''+p(t)y'+[\lambda^2r(t)-q(t)]y=R(t), \]

where \(t=1\pm x\).

The asymptotic representations for the coefficients \(\widetilde y_n(x)\) entering equation (2) are computed in paper (6), namely:

\[ S_n=\frac{\beta-\alpha-\gamma x_0}{2}+O(n^{-1}),\qquad A(n)=O(n^{-1}), \]

\[ D_n= \begin{cases} \gamma(x_0^2-1)[1+o(1)], & \widetilde y_n(x_0)=0,\\ o(1), & \widetilde y_n(x_0)\ne 0. \end{cases} \]

The scheme considered in paper (4) is stable with respect to violation of homogeneity of the indicated type for the values \(-1<\alpha,\beta\le +1\), \(0\le\gamma<2\). In this case the integrals additionally arising in the remainder term are convergent, and their order, owing to \(A(n)\), is at least \(n^{-2}\). As for the values \(\alpha,\beta>+1\), \(\gamma\ge 2\), the polynomials obtained in this case are expressed in finite form through polynomials for which \(-1<\alpha,\beta\le +1\), \(0\le\gamma<2\) (see (1), p. 28). Taking all this into account and referring to representation (16) of paper (4), we write out the following asymptotic formulas:

\[ \tilde y_n(x) \simeq C_1 |x-x_0|^{-\frac{\gamma}{2}}(1-x)^{-\frac{2\alpha+1}{4}}(1+x)^{-\frac{2\beta+1}{4}}(1-x_0)^{\frac{\alpha+1}{2}}(1+x_0)^{\frac{\beta+1}{2}} \]
\[ {}\times \sqrt{|\arcsin x-\arcsin x_0|}\, J_\nu\!\left(\sqrt{n(n+1+\alpha+\beta+\gamma)}\,|\arcsin x-\arcsin x_0|\right), \tag{3} \]

where \(-1<l_1 \le x \le x_0\) or \(x_0 \le x \le l_2 < +1\),

\[ \nu=\pm \frac{\sqrt{(\gamma-1)^2+4q(x_0)}}{2}, \qquad q(x_0)=\frac{D_n}{x_0^2-1}, \]

and the sign is chosen depending on the order of the zero of \(\tilde y_n(x)\) at the point \(x=x_0\), i.e., here one may consider separately the asymptotic representations of polynomials that have, for \(x=x_0\), a root for all sufficiently large \(n\) \(\bigl(D_n=\gamma(x_0^2-1)[1+o(1)]\bigr)\), and of polynomials that do not vanish for \(x=x_0\) \(\bigl(D_n=o(1)\bigr)\). It is easy to see that \(\nu \simeq \frac{\gamma+1}{2}\) and \(\nu \simeq \frac{\gamma-1}{2}\), respectively.

Further, for \(-1 \le x \le l_1 < x_0\) we obtain, putting \(\nu=\beta\),

\[ \tilde y_n(x) \simeq C_2(1-x)^{-\frac{2\alpha+1}{4}}(1+x)^{-\frac{2\beta+1}{4}}(x_0-x)^{-\frac{\gamma}{2}}(1+x_0)^{\frac{\gamma}{2}} \]
\[ {}\times \sqrt{\frac{\pi}{2}+\arcsin x}\, J_\nu\!\left(\sqrt{n(n+1+\alpha+\beta+\gamma)}\left(\frac{\pi}{2}+\arcsin x\right)\right), \tag{4} \]

and for \(x_0<l_2 \le x \le +1\) we shall have, for \(\nu=\alpha\),

\[ \tilde y_n(x) \simeq C_3(1-x)^{-\frac{2\alpha+1}{4}}(1+x)^{-\frac{2\beta+1}{4}}(x-x_0)^{-\frac{\gamma}{2}}(1-x_0)^{\frac{\gamma}{2}} \]
\[ {}\times \sqrt{\frac{\pi}{2}-\arcsin x}\, J_\nu\!\left(\sqrt{n(n+1+\alpha+\beta+\gamma)}\left(\frac{\pi}{2}-\arcsin x\right)\right). \tag{5} \]

If one uses the asymptotic representations of the Bessel functions \(J_\nu(z)\) as \(z\to 0\) and \(z\to\infty\), then one may conclude that the orthonormal polynomials

\[ \hat y(x)=a_n\bigl(x^n-S_nx^{n-1}+d_nx^{n-2}+\cdots\bigr),\qquad \gamma\ge 0, \]

at the point \(x=x_0\) have order \(n^{\gamma/2}\), and at the endpoints of the interval of orthogonality, respectively, \(n^{\beta+1/2}\) and \(n^{\alpha+1/2}\) \((^5,^6)\).

Saratov State
Pedagogical Institute

Received
29 VI 1961

References

\(^1\) G. Szegő, Am. Math. Soc. Colloq. Publ., 23 (1939).
\(^2\) S. N. Bernstein, On polynomials orthogonal on a finite interval, Kharkov, 1937.
\(^3\) G. Szegő, J. Industr. and Appl. Math., 7, No. 3, 311 (1959).
\(^4\) V. P. Konoplev, DAN, 118, No. 1, 25 (1958).
\(^5\) G. N. Watson, Theory of Bessel Functions, Part 1, IL, 1949.
\(^6\) J. Shohat, Trans. Am. Math. Soc., 29, No. 3, 569 (1927).