Abstract
The representation of analytic functions as convergent series in Jacobi polynomials is reformulated using the Hadamard principal part of integrals for all . The coefficients of the series are given as usual integrals in the classical case (when ) or by their Hadamard principal part when they diverge. As an application it is shown that nonhomogeneous differential equations of hypergeometric type do generically have a unique solution which is analytic at both singular points in .
1. Introduction
The purpose of this note is to provide a unified and easy-to-use formalism of expansion of analytic functions in Jacobi series for general parameters. Such expansions are useful in approximation theory, as well as in a variety of other areas, such as the study of differential equations (see, e.g., [1–3]) and non-self-adjoint spectral problems which appear in the study of stability of nonlinear partial differential equations.
This formalism is applied to answer an open question on differential equations in Section 2.
1.1. An Overview of Jacobi Polynomials for Classical Parameters
We summarize below a few well known facts about Jacobi polynomials for (see [4] or [5] for an overview).
The Jacobi polynomials are determined by the condition of mutual orthogonality on the interval with respect to the weightnamely,whereRelation (2) and the condition that the coefficient of the leading term, in , is positive uniquely determine the Jacobi polynomials.
In the particular case when these polynomials are called Gegenbauer (or ultraspherical), and when they are called Chebyshev polynomials.
As any orthogonal polynomials, Jacobi polynomials satisfy a three term recurrence relation where , and
Also, the Jacobi polynomials satisfy the Rodrigues formula: where
Furthermore, satisfy the differential equation:i.e., are eigenfunctions of a self-adjoint operator.
It is a classic result that analytic functions can be expanded in Jacobi series (see [6] p.245).
Theorem 1. Let . Let be analytic in the interior of an ellipse with foci at . Denote by the greatest such ellipse. Then has an expansionwhich is convergent in the interior of and divergent outside.
Of course, due to the orthogonality relation (2) the coefficients in (9) can be calculated as
1.2. Jacobi Polynomials for General Parameters
For with all the results presented above still hold, except that now are eigenfunctions of a non-self-adjoint operator.
For other complex values of , however, the integrals in (2) and in (10) diverge and the Jacobi polynomials can no longer be thought of being an orthogonal system. Nevertheless, Jacobi polynomials can be defined through the Rodrigues formula, or, most importantly, through the three-terms recurrence. The latter then implies, by Favard’s Theorem, that there is a bilinear form with respect to which the polynomials are orthogonal [7, 8].
Many decades after Theorem 1 has been established, in his 1974 paper [9], Carlson showed that expansions (9) hold for general complex parameters provided that . The formulas in [9] use an integral kernel and are not easy to use in practice (see Theorem 4 in Section 4.3). More recently Kuijlaars, Martínez-Finkelshtein, and Orive find by analytic continuation that orthogonality of Jacobi polynomials can be established by integration on special paths in the complex plane; they also derive an associated Riemann-Hilbert problem; in some cases incomplete or quasi-orthogonality or even multiple orthogonality conditions are found [10, 11].
In the monograph [12] it is shown that, for general parameters, there exists a weight with respect to which the Jacobi polynomials are orthogonal along closed paths in the complex plane (provided that are not negative integers or ). Also, the coefficients of the Jacobi expansion of an analytic function are expressed by weighted integrals of the function along ellipses in the complex plane, the weights being the associated Jacobi polynomials .
Later, in [13], it was shown that for with the Jacobi polynomials do satisfy that (2) only now the integral should be understood as its Hadamard finite part: (2) taken in the Hadamard finite part is the bilinear form with respect to which the Jacobi polynomials are orthogonal. It was also shown in [13] that the Hadamard finite part of the integral is the analytic continuation in of the bilinear form (2) for the cases when the integral no longer converges.
1.3. Jacobi Series for General Parameters through the Hadamard Finite Part
We reformulate both the classical Theorem 1 and the generalized ones, Carlson’s result, and the one in [12], from a unified point of view, easy to use in applications: Theorem 2 states that even in the case of general complex parameters (2) and (10) are valid provided that possible divergent integrals are replaced by their Hadamard finite part. A brief introduction to the Hadamard finite part is found in Appendix.
The result is formulated for Jacobi polynomials on the interval .
Theorem 2. Let .
Consider the (nonnormalized) Jacobi polynomials on given by the Rodrigues formula:(i) Orthogonality. The polynomials are orthogonal with respect to a bilinear form: where denotes the Hadamard finite part of the integral and (ii) Completeness. Let be the interior of an ellipse with foci and , and let be analytic in . Then has the expansionwhereExpansion (14) converges absolutely and uniformly on every compact set in .
Since it was established in [13] that analytic continuation in of integrals of the type (2) and (10) is given by their Hadamard finite part, then Theorem 2 follows by analytic continuation of Theorem 1. We include however a careful proof in Appendix, Section 4.
2. Analytic Solutions of Inhomogeneous Hypergeometric Equations
In this section we use Theorem 2 to investigate existence of eigenfunctions for some non-self-adjoint operators; more precisely, we establish existence of unique analytic solutions of inhomogeneous differential equations of hypergeometric type under precise conditions on the parameters.
The differential equation of hypergeometric typehas solutions which are analytic at the singular point and solutions analytic at (as it is well known, and easy to establish using Frobenius theory [14]). But, for generic parameters no nonzero solution is analytic at both singular points and .
We investigate if an inhomogeneous version, (17) below, has solutions that are analytic at both and . This type of questions appear as generalized eigenfunction problems in many types of applications, for instance, in the study of blow-up of wave maps [15, 16].
We find the answer to be positive.
Theorem 3. Consider the following differential equation of hypergeometric type with a nonhomogeneous term:where is a function that is analytic in the domain which is the interior of an ellipse with foci at .
Assume that andThen (17) has a unique solution which is analytic at both and , and therefore it is analytic on the whole domain where is analytic.
Proof. Let be the Jacobi polynomials on defined by (11) with .
DenoteThen by (43) each polynomial satisfieswhich implies thatso that is a solution of (17) for the case when .
We are thus led to the idea of expanding the inhomogeneous term in a series in Jacobi polynomials.
By Theorem 2 there exist constants so that with the series converging uniformly on every compact contained in and given by the formula We look for a solution of (17) which is analytic at both and , and therefore it is analytic in ; also by Theorem 2, such a solution has an expansion which plugged in (17) yieldswhich by (21) impliesand using (12) it follows that for all . In view of (18) and (19) it follows thatWe thus found a unique solution analytic in of (17) as the sum of the Jacobi serieswith given by (22). The series is obviously convergent everywhere the series of is.
3. Appendix: The Hadamard Finite Part
The concept of the finite part of a (possibly divergent) integral was introduced by Hadamard [17] as a convenient way to express solutions of differential equations. He showed that this finite part of an integral (which coincides with the usual value if the integral is convergent) can be combined and manipulated in much the same way as usual integrals: they are additive on the interval of integration, changes of variable are allowed, etc. (They do not behave well with respect to inequalities.) The finite part can be calculated either by Taylor series or by integration along closed paths in the complex plane.
Subsequently the Hadamard finite part has been interpreted in terms of distributions (see, e.g., [18]) and it turned out that many problems of mathematical physics have solutions expressible as the Hadamard finite part of (divergent) integrals, and numerical methods of calculations have been subsequently developed (see, for example, [19]).
The present section contains some properties of the Hadamard finite part of integrals of the type with analytic at ; when , , its Hadamard finite part is denoted here by
Consider a function analytic at : a series with nonzero radius of convergence . The integralconverges for and for we haveand we see that the final result is analytic at . For all other complex values of , the Hadamard finite part of the integral is defined by
This definition represents the analytic continuation in of the function (28) to in the complex plane; when is a negative integer the continuation has poles of order one.
Remark. For practical calculations one does not need to expand in its full power series, just a finite number of terms are needed. For example, to calculatefirst write where is analytic at . Thenwhere the first term isand the last term is an usual integral.
4. Appendix: Proof of Theorem 2
4.1. Jacobi Polynomials on
Substitutingin the Rodrigues formula (6) we obtain (11) which shows that the polynomials defined by (11) are indeed Jacobi polynomials (up to multiplicative constants).
It is easy to see that the leading coefficient of is the coefficient of which equals(Note that if is a negative integer these polynomials are no longer a complete system.)
It is also easy to see that depend polynomially on and .
4.2. Orthogonality for General Parameters
Formula (12) was proved in [13]. But one can also reason that it follows by analytic continuation, in the following way. For substituting (34) in (2) we obtaina relation which, by analytic continuation, then holds for all other complex values of , implying (12).
Formula (13) can be obtained by a straightforward calculation based on (34) and (2) and (3).
4.3. Completeness
As it was mentioned, Theorem 2(ii) follows by analytic continuation of Theorem 1.
We will however give a careful proof based on the result in [9], which is the following.
Theorem 4 (Theorem 1.1 in [9] for ). Let be an open elliptic disk with foci and , and let be analytic on . Let and assume . Then, for every ,where are scalar multiples of , given by the formula wherefor and its analytic continuation for other complex values of , and where where is a rectifiable Jordan curve in which encircles the segment in the positive direction.
The series (37) converges absolutely on , uniformly on every compact set in .
A straightforward calculation can be also made to link (14) and (37), yieldingandwhere are given by (38); denotes the rising factorial: . We illustrate below the main steps of the calculation.
In [9] is shown to be a multiple of a Jacobi polynomial by using the binomial formula in (38), then integrating term by term (this is also correct if one uses the Hadamard principal part in case the integrals diverge). Then one compares with the expression of obtained by using Leibniz’s rule to expand its Rodrigues formula.
Integration by parts also holds for the Hadamard finite part, and were proved to be an orthogonal set in [13]. It only remains to calculate the “norm” of in this sense.
We first calculate the dominant term in : it is found by retaining only the highest powers of in its Rodrigues formula, yielding: where .
Then, using the fact that is orthogonal to all the polynomials of degree less than and then using the Rodrigues formula and integrating by parts times we obtain
We will need the differential equation (8) transcribed for the Jacobi polynomials on .
Proposition 5. The Jacobi polynomials defined by (11) for general satisfy the differential equation
Proof. The proof is a straightforward calculation using (34) in (8).
Data Availability
The data used to support the findings of this study are included within the article
Conflicts of Interest
There are no conflicts of interest regarding the publication of this paper.