Orthogonality Properties of the Pseudo-Chebyshev Functions ( Variations on a Chebyshev ’ s Theme )

The third and fourth pseudo-Chebyshev irrational functions of half-integer degree are defined. Their definitions are connected to those of the firstand second-kind pseudo-Chebyshev functions. Their orthogonality properties are shown, with respect to classical weights.


Introduction
In a recent article [1], starting from the complex Bernoulli spiral, the sets of classical Chebyshev polynomials of the first-and second-kind have been extended to the case of fractional indices.The resulting functions have been called pseudo-Chebyshev polynomials (or functions), since actually, they are not polynomials, but irrational functions.However, in the particular case of half-integer indices, the functions T n+1/2 , U n+1/2 satisfy the same properties of their classical counterparts, including recurrence relations, differential equations, and orthogonality properties, as has been proven in [2].
In this article, after recalling the definitions of the third-and fourth-kind Chebyshev polynomials, we introduce the third-and fourth-kind pseudo-Chebyshev functions V n+1/2 , W n+1/2 , focusing our attention on the orthogonality properties satisfied by these new irrational functions.
Even though there are links to the classical Chebyshev polynomials, it seems unusual to find a set of irrational functions that satisfy properties so similar to their polynomial counterparts.
A clear possible application of the introduced functions is the expansion of irrational functions in non-trigonometric Fourier series, and-for the third-kind pseudo-Chebyshev functions-the construction of quadrature rules applied to functions whose singularities occur only at the end points (+1 and −1) (see [3] for similar situations).In Section 8, the connection with the Dirichlet kernel is shown, and in a forthcoming article, further properties derived from links to classical Chebyshev polynomials will be highlighted.

Chebyshev Polynomials
The Chebyshev polynomials of the first-and second-kind were introduced by Pafnuty L. Chebyshev in the 19th Century.They can be derived as the real and imaginary part of the exponential function e i n θ = (cos θ + i sin θ) n , setting x = cos θ and using the Euler formula (see [4] for details).
The first-kind Chebyshev polynomials are important in approximation theory and Gaussian quadrature rules.Indeed, by using their roots-called Chebyshev nodes-the resulting interpolation polynomial minimizes the Runge phenomenon.Furthermore, the relevant approximation is the best approximation to a continuous function under the maximum norm.
Linked with such polynomials are the Chebyshev polynomials of the second-kind, which appear in computing the powers of 2 × 2 non-singular matrices [5].Generalizations of these polynomials have been also introduced, in particular for computing powers of higher order matrices (see, e.g., [6,7]).
An excellent book is [3].The importance of these polynomial sets in applications is shown in [8].
It is also useful to notice that Chebyshev polynomials represent an important tool to derive integral representations [9,10], and that they can be generalized by using the properties and formalism of the Hermite polynomials [11], for instance by introducing multi-variable polynomials recognized as belonging to the Chebyshev family [12][13][14].
Recently, the Chebyshev polynomials of the first-and second-kind have been used in order to represent the real and imaginary part of complex Appell polynomials [15].
Other sets of orthogonal polynomials, linked to the above-mentioned ones, are known as the third-and fourth-kind Chebyshev polynomials.
The third-and fourth-kind Chebyshev polynomials have been studied and applied by several scholars (see, e.g., [16][17][18]), because they are useful in quadrature rules, when the singularities occur only at one of the end points (+1 or − 1) (see [3]).Furthermore, they have been recently applied in numerical analysis to solve high odd-order boundary-value problems with homogeneous or nonhomogeneous boundary conditions [17].

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The third-kind Chebyshev polynomials (Figure 3) can be expressed in terms of the first-kind pseudo-Chebyshev functions as follows:

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The fourth-kind Chebyshev polynomials (Figure 4) can be expressed in terms of the second-kind pseudo-Chebyshev functions as follows: The third-and fourth-kind Chebyshev polynomials satisfy the orthogonality property: 1

Third-and Fourth-Kind Pseudo-Chebyshev Functions
In what follows, we introduce the third-and fourth-kind pseudo-Chebyshev functions.

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The third-kind pseudo-Chebyshev functions (Figure 5) are linked to the first-kind pseudo-Chebyshev functions, by means of the equation:

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The fourth-kind pseudo-Chebyshev functions (Figure 6) are linked to the second-kind pseudo-Chebyshev functions, by means of the equation:

Recurrence Relations
Note that: and therefore, Equation ( 8) can be written as: according to the symmetry property . Furthermore, we have: that is, Theorem 1.The first-kind pseudo-Chebyshev functions satisfy the recurrence relation: Proof of Theorem 1.We use induction.Equation ( 11) holds for k = 1, according to Equation (9).Recalling Equation (10), the induction hypothesis is written: Therefore, using the recursions of classical Chebyshev polynomials, we have: In a similar way, the recurrence relation of the second-kind pseudo-Chebyshev functions can be proven [1,2].
Theorem 2. The pseudo-Chebyshev functions V k+1/2 (x), W k+1/2 (x) verify the same recurrence relation of the classical Chebyshev polynomials, with suitable initial conditions.More precisely, we have: Proof of Theorem 2. Note that third-and fourth-kind pseudo-Chebyshev functions are defined multiplying the pseudo-Chebyshev functions (x) by functions that are independent of k.Therefore, they must satisfy the same recurrence relation of classical Chebyshev polynomials.Furthermore, their initial conditions can be easily derived from their definitions, taking into account the symmetry properties of the circular functions.