EXPONENTIAL POLYNOMIALS AND THE DECOMPOSITION OF SOME STANDARD SIGNALS

V. M. OSIPOV

(Presented by the scientific seminar of the Departments of Automation and Telemechanics and Automatic Systems)

As is known [3, 6, 8], the eigenfunctions of a self-adjoint Sturm–Liouville differential operator with self-adjoint boundary conditions form a system of functions orthogonal with respect to a certain weight, complete in the space \(L^2 \rho(x)\), i.e.

\[ \int_a^b \rho(x) U_i(x) U_k(x)\,dx \begin{array}{ll} =0 & i \ne k,\\ \ne 0 & i = k, \end{array} \]

where \(\rho(x)\) is the weight function, and \(U_i(x)\) and \(U_k(x)\) are the eigenfunctions corresponding to the eigenvalues \(\lambda_i\) and \(\lambda_k\). In particular, the hypergeometric function

\[ F(\alpha;\,\beta;\,\gamma;\,x) = \sum_{\kappa=0}^{\infty} \frac{(\alpha)_\kappa(\beta)_\kappa}{(\gamma)_\kappa\,\kappa!} x^\kappa \tag{1} \]

for a definite relation among the constants is a solution of the Sturm–Liouville problem for the differential equation whose self-adjoint form is

\[ \frac{d}{dx} \left[ x^\gamma(1-x)^{\alpha+\beta+1-\gamma} U' \right] - \alpha\beta x^{\gamma-1}(1-x)^{\alpha+\beta-\gamma} U = 0. \tag{2} \]

This equation, called hypergeometric, is of considerable interest for applications.

If the parameter \(\alpha\) or \(\beta\) is a negative integer (or zero), then series (1) terminates and becomes a polynomial of the corresponding degree. This circumstance, if one considers the interval \((0,1)\), leads to self-adjoint boundary conditions for

\[ \gamma > 0;\quad \alpha+\beta-1-\gamma > 0. \tag{3} \]

If we put

\[ \alpha=-n\;(n=0,1,2\ldots);\quad \beta=n+\gamma+\delta-1, \]

then the hypergeometric function forms a sequence of polynomials

\[ P_n^{\gamma\delta}(x) = F(-n,\,n+\gamma+\delta-1;\,\gamma;\,x) \quad (n=0,1,2\ldots), \tag{4} \]

which are called Jacobi polynomials [4,6,8].

For any \(\gamma\) and \(\delta\) satisfying conditions (3), the Jacobi polynomials are orthogonal with weight

\[ \rho(x)=x^{\gamma-1}(1-x)^{\delta-1}\quad 0\leqslant x\leqslant 1. \tag{5} \]

Introduce the substitution \(x=e^{-at}\), which transforms the interval \((0,1)\) into the interval \((0,\infty)\) for the new variable \(t\). As a result, the sequence (4) is transformed into a new sequence of polynomials, but now with respect to \(e^{-at}\). These are exponential Jacobi polynomials, or, for short, “\(e\)”-polynomials. We shall denote them by the former symbol, but instead of the argument \(x\) we shall write \(t\).

\[ P_n^{(\gamma,\delta)}(e^{-at}) \equiv P_n^{(\gamma,\delta)}(t) =F(-n,n+\gamma+\delta-1;\gamma;e^{-at}). \tag{6} \]

The parameter “\(a\)” is real and positive. It is easy to verify that the Jacobi “\(e\)”-polynomials are orthogonal on the interval \((0,\infty)\) with respect to the weight

\[ W(t)=e^{-\gamma at}(1-e^{-at})^{\delta-1}. \tag{7} \]

Of greatest interest are the following particular values of \(\gamma\) and \(\delta\):

\[ 1.\quad \gamma=\delta=\frac{1}{2};\quad W(t)=e^{-\frac{at}{2}}(1-e^{-at})^{-\frac{1}{2}}, \]

\[ P_n^{(\frac12,\frac12)}(t)=T_n^*(t) =F\left(-n,n;\frac12;e^{-at}\right). \tag{8} \]

We obtain the Chebyshev “\(e\)”-polynomials of the first kind.

\[ 2.\quad \gamma=\delta=1;\quad W(t)=e^{-at}, \]

\[ P_n^{(1,1)}(t)=P_n^*(t)=F(-n,n+1;1;e^{-at}). \tag{9} \]

This case gives the Legendre “\(e\)”-polynomials.

\[ 3.\quad \gamma=\delta=\frac{3}{2};\quad W(t)=e^{-\frac{3}{2}at}(1-e^{-at})^{\frac12}, \]

\[ P_n^{(\frac32,\frac32)}(t)=U_n^*(t) =F\left(-n,n+2;\frac32;e^{-at}\right), \tag{10} \]

In this case we have the Chebyshev “\(e\)”-polynomials of the second kind.

1. Expansion of Some Functions in Series of Legendre “\(e\)”-Polynomials

The exponential Legendre polynomials satisfy the following differential equation:

\[ \frac{d}{dt}\left[(1-e^{-at})\frac{dP_n^*(t)}{dt}\right] +n(n+1)a^2e^{-at}P_n^*(t)=0. \tag{1-1} \]

Their explicit expression is obtained immediately by expanding the hypergeometric series \([1, 7]\):

\[ P_n^*(t)=F(-n,n+1;1;e^{-at}) =\sum_{\kappa=0}^{n}(-1)^\kappa \frac{(n+\kappa)!}{(\kappa!)^2(n-\kappa)!}e^{-\kappa at}, \tag{1-2} \]

\[ P_0^*(t)=1;\quad P_1^*(t)=1-2e^{-at};\quad P_2^*(t)=1-6e^{-at}+6e^{-2at}; \]

\[ P_3^*(t)=1-12e^{-at}+30e^{-2at}-20e^{-3at}\ \text{and so on.} \]

Let us note that the values of the coefficients of the series (1–2) coincide with the corresponding values of the coefficients of the shifted Legendre polynomials defined on the interval \((0,1)\).

These coefficients are given in [6] up to and including \(n=14\). Let us also note that the Legendre “\(e\)-polynomials” can be obtained from the ordinary polynomials defined on the interval \((-1,1)\), if one makes the substitution \(x \to 1-2e^{-at}\) and carries out the necessary transformations. Thus, all relations derived for ordinary polynomials remain valid also for the “\(e\)-polynomials,” if the indicated substitution is made. It should be borne in mind, however, that

\[ \frac{dP_n(x)}{dx}=\frac{dP_n^*(t)}{dt}\cdot\frac{e^{at}}{2a}. \]

The recurrence relation connecting three successive “\(e\)-polynomials” has the form [7]

\[ (n+1)P_{n+1}^*(t)-(2n+1)(1-2e^{-at})P_n^*(t)+nP_{n-1}^*(t)=0. \tag{1-3} \]

Further, one can obtain

\[ \frac{dP_{n+1}^*(t)}{dt}-\frac{dP_{n-1}^*(t)}{dt}=(2n+1)\,2ae^{-at}P_n^*(t). \tag{1-4} \]

The following relations also hold:

\[ P_n^*(0)=(-1)^n;\quad P_n^*(\infty)=1, \tag{1-5} \]

\[ |P_n^*(t)|\leqslant 1 \quad 0\leqslant t\leqslant \infty. \tag{1-6} \]

Let us consider the question of expanding an arbitrary function defined on the interval \((0,\infty)\) in a series in Legendre “\(e\)-polynomials.” Suppose that the expansion

\[ f(t)=\sum_{n=0}^{\infty} c_n P_n^*(t) \tag{1-7} \]

holds.

The convergence of this series is established by the following theorem:

“If \(f(t)\) is a piecewise-continuous function satisfying the condition

\[ \int_{0}^{\infty} e^{-at}[f(t)]^2\,dt<\infty, \tag{1-8} \]

then the expansion (1–7) with coefficients

\[ c_n=a(2n+1)\int_{0}^{\infty} e^{-at} f(t)P_n^*(t)\,dt \tag{1-9} \]

converges to \(f(t)\) at every interior point of the interval \((0,\infty)\) that is a point of continuity of this function.” This theorem is an analogue of the well-known theorem on the expansion of an arbitrary function defined on the interval \((-1,1)\) in a series in ordinary Legendre polynomials [7]. The expansion coefficients (1–9) can readily be obtained from the given Laplace transform of the function \(f(t)\).

Let \(F(p)\) be the Laplace transform of the function \(f(t)\), i.e.,

\[ F(p)=\int_{0}^{\infty} e^{-pt}f(t)\,dt, \]

then, taking (1–2) into account, we can write

\[ \begin{aligned} c_n &=a(2n+1)\sum_{k=0}^{n}(-1)^k \frac{(n+k)!}{(k!)^2(n-k)!} \int_{0}^{\infty} e^{-(k+1)at} f(t)\,dt \\ &=a(2n+1)\sum_{k=0}^{n}(-1)^k \frac{(n+k)!}{(k!)^2(n-k)!}\,F[(k+1)a], \end{aligned} \tag{1-10} \]

where \(F[(k+1)a]\) is the Laplace transform of the function \(f(t)\), in which \(p\) has been replaced by \((k+1)a\). This circumstance makes it possible to perform the inversion of the Laplace transform in the form of a series in Legendre “\(e\)-polynomials.”

Let us note that, since the system of Legendre “\(e\)-polynomials” is complete in the space \(L^2_{w(t)}[W(t)=e^{-at}]\), Parseval’s equality [8] holds:

\[ \int_{0}^{\infty} e^{-at}|f(t)|^2\,dt = \sum_{n=0}^{\infty}\frac{c_n^2}{(2n+1)a}, \tag{1-11} \]

since

\[ \int_{0}^{\infty} e^{-at}[P_n^*(t)]^2\,dt = \frac{1}{(2n+1)a}. \tag{1-12} \]

Let us find the Laplace transform of the “\(e\)-polynomial” \(P_n^*(t)\). We have

\[ P_n(p)=\int_{0}^{\infty}e^{-pt}P_n^*(t)\,dt = \int_{0}^{\infty}e^{-pt}F(-n,n+1;1;e^{-at})\,dt. \tag{1-13} \]

If we make the substitution \(x=e^{-at}\), we obtain the tabulated integral [4]

\[ \begin{aligned} P_n(P) &=\frac{1}{a}\int_{0}^{1} x^{\frac{p}{a}-1}F(-n,n+1;1;x)\,dx \\ &= \frac{\Gamma\!\left(\frac{p}{a}\right)\cdot \Gamma\!\left(1+n-\frac{p}{a}\right)} {a\Gamma\!\left(1-\frac{p}{a}\right)\cdot \Gamma\!\left(1+n+\frac{p}{a}\right)}. \end{aligned} \]

Since

\[ \frac{\Gamma\!\left(1+n-\frac{p}{a}\right)} {\Gamma\!\left(1+n+\frac{p}{a}\right)} = \frac{ \left(n-\frac{p}{a}\right) \left(n-1-\frac{p}{a}\right)\cdots \left(1-\frac{p}{a}\right) \Gamma\!\left(1-\frac{p}{a}\right) }{ \left(n+\frac{p}{a}\right) \left(n-1+\frac{p}{a}\right)\cdots \left(1+\frac{p}{a}\right) \frac{p}{a}\Gamma\!\left(\frac{p}{a}\right) }, \]

we finally obtain

\[ P_n(p)=\frac{1}{p}\prod_{k=0}^{n-1} \frac{(k+1)a-p}{(k+1)a+p}\doteq P_n^*(t), \tag{1-13} \]

Let us now consider the expansion of the exponential function

\[ f(t)=e^{-pt}, \tag{1-14} \]

(where \(p\) is a complex parameter) in a series in Legendre “\(e\)-polynomials,”

which generates a large number of other useful decompositions. Condition (1-8) will be satisfied if \(Rep > -\dfrac{a}{2}\). Let us find the coefficients \(D_n(p)\) of the series

\[ e^{-pt}=\sum_{n=0}^{\infty}D_n(p)P_n(t). \tag{1-15} \]

According to the general formula (1-9), we have

\[ D_n(p)=a(2n+1)\int_0^\infty e^{-at}e^{-pt}P_n^*(t)\,dt. \]

Comparing with (1-13), we find that

\[ D_n(p)=a(2n+1)P_n(p+a)=\frac{(2n+1)a}{p+a}\prod_{\kappa=0}^{n-1}\frac{\kappa a-p}{(2+\kappa)a+p}. \tag{1-16} \]

Thus,

\[ e^{-pt}=\sum_{n=0}^{\infty}\frac{(2n+1)a}{p+a}\prod_{\kappa=0}^{n-1}\frac{\kappa a-p}{(2+\kappa)a+p}\,P_n^*(t). \tag{1-17} \]

Let us note that the series converges uniformly if \(Rep>-\dfrac{a}{2}\). Suppose that \(p\) is a real quantity and, in particular, a multiple of “\(a\),” i.e. \(p=ma\) \((m=1,2,\ldots)\); then expansion (1-17) terminates at the \((m+1)\)-st term and takes the form

\[ e^{-mat}=\sum_{n=0}^{m}\frac{2n+1}{m+1}\prod_{\kappa=0}^{n-1}\frac{\kappa-m}{2+\kappa+m}\,P_n^*(t). \tag{1-18} \]

This formula makes it possible to express powers of \(e^{-at}\) in the form of a linear combination of a finite number of “\(e\)”-polynomials. For the first few values of “\(m\)” we shall have

\[ \left. \begin{aligned} e^{-at}&=\frac{1}{2}-\frac{1}{2}P_1^*(t),\\ e^{-2at}&=\frac{1}{3}-\frac{1}{2}P_1^*(t)+\frac{1}{6}P_2^*(t),\\ e^{-3at}&=\frac{1}{4}-\frac{9}{20}P_1^*(t)+\frac{1}{4}P_2^*(t)-\frac{1}{20}P_3^*(t). \end{aligned} \right\} \tag{1-19} \]

Let now \(p=a+jb\); then, taking into account that

\[ e^{-at}\cos bt=Re\,e^{-(a+jb)t}, \]

\[ e^{-at}\sin bt=-\,Im\,e^{-(a+jb)t}, \]

we obtain

\[ \left. \begin{aligned} e^{-at}\cos bt &=Re\sum_{n=0}^{\infty}\frac{(2n+1)a}{2a+jb} \prod_{\kappa=0}^{n-1}\frac{(\kappa-1)a-jb}{(3+\kappa)a+jb}\,P_n^*(t),\\ e^{-at}\sin bt &=-\,Im\sum_{n=0}^{\infty}\frac{(2n+1)a}{2a+jb} \prod_{\kappa=0}^{n-1}\frac{(\kappa-1)a-jb}{(3+\kappa)a+jb}\,P_n^*(t). \end{aligned} \right\} \tag{1-20} \]

These expansions converge the faster, the smaller \(\dfrac{b}{a}\) is. From the decompo-

relations (1–5) one can obtain an expansion for the function \(f(t)=at\). Obviously,

\[ t=-\left.\frac{d}{dp}\left(e^{-pt}\right)\right|_{p=0} \tag{1-21} \]

and, consequently, taking into account the uniform convergence of the series (1-15),

\[ t=-\left.\sum_{n=0}^{\infty}D'_n(p)P_n^*(t)\right|_{p=0}. \tag{1-22} \]

According to (1-16), \(D_n(p)\) can be represented in the form

\[ D_0(p)=\frac{a}{p+a}, \]

\[ D_n(p)=-\frac{(2n+1)ap}{(p+a)(p+2a)}\,N_n(p),\quad n=1,2\ldots, \]

where

\[ N_n(p)=\prod_{\kappa=1}^{n-1}\frac{\kappa a-p}{(2+\kappa)a+p}\quad [N_1(p)=1]. \]

The derivatives will be equal to

\[ D'_0(p)=-\frac{a}{(p+a)^2}, \]

\[ D'_n(p)=-(2n+1)a\frac{2a^2-p^2}{(p+a)^2(p+2a)^2}\,N_n(p) -\frac{(2n+1)ap}{(p+a)(p+2a)}\,N'_n(p). \]

Putting \(p=0\), we shall have \([N'_n(0)\ne\infty]\):

\[ D'_0(0)=-\frac{1}{a};\quad D'_n(0)=-\frac{2n+1}{2a}\,N_n(0) \]

\[ N_n(0)=\prod_{\kappa=1}^{n-1}\frac{\kappa}{2+\kappa} =\frac{1\cdot2\cdot3\ldots(n-1)}{3\cdot4\ldots(n-1)n(n+1)} =\frac{2}{n(n+1)}. \]

Finally we obtain

\[ D'_n(p)\big|_{p=0}=D'_n(0)=-\frac{2n+1}{an(n+1)}, \]

The desired expansion has the form

\[ at=1+\sum_{n=1}^{\infty}\frac{2n+1}{n(n+1)}P_n^*(t);\quad 0\le t<\infty. \tag{1-23} \]

Similarly one can obtain an expansion for \(te^{-at}\), since

\[ te^{-at}=-\left.\frac{d}{dP}\left(e^{-pt}\right)\right|_{p=a}. \]

It is simpler, however, to use the rule for multiplying a series by an exponential factor. Let

\[ f(t)=\sum_{n=0}^{\infty}c_nP_n^*(t). \]

Let us find the expansion for the function \(e^{-at}f(t)\). We have

\[ e^{-at}f(t)=\sum_{n=0}^{\infty}c_ne^{-at}P_n^*(t). \]

From the basic recurrence relation (1-3) we find

\[ e^{-at}P_n^*(t)=\frac{1}{2}P_n^*(t)-\frac{n}{2(2n+1)}P_{n-1}^*(t)-\frac{n+1}{2(2n+1)}P_{n+1}^*(t). \]

Substituting this expression into the series and carrying out the transformation of indices, we obtain

\[ e^{-at}f(t)=\frac{1}{2}\left(c_0-\frac{1}{3}c_1\right)+ \frac{1}{2}\sum_{n=1}^{\infty}\left[ c_n-\frac{n+1}{2n+3}c_{n+1}- \frac{n}{2n-1}c_{n-1}\right]P_n^*(t). \tag{1-24} \]

Multiplying (1-23) by \(e^{-at}\) and using formula (1-24), as a result we shall have the desired expansion

\[ ate^{-at}=\frac{1}{4}+\frac{1}{12}P_1^*(t)- \sum_{n=2}^{\infty}\frac{2n+1}{(n-1)n(n+1)(n+2)}P_n^*(t). \tag{1-25} \]

The series converges rather rapidly.

Let us consider the expansion of some discontinuous functions that are widely used in engineering applications as standard inputs.

We return to expansion (1-15). Fix the quantity \(t\), putting \(t=\tau\); then we shall have

\[ e^{-p\tau}=\sum_{n=0}^{\infty}D_n(p)P_n^*(\tau). \tag{1-26} \]

If (1-26) is regarded as the result of the Laplace transform of some time relation, then the latter can be found by inverting both sides of (1-26). On the left, inversion gives the Dirac delta function, i.e.

\[ e^{-p\tau}\doteq \delta(t-\tau). \]

Taking (1-13) and (1-16) into account, we arrive at the correspondence

\[ D_n(p)\doteq a(2n+1)e^{-at}P_n^*(t). \tag{1-27} \]

By virtue of the uniqueness of the Laplace transform we obtain

\[ \delta(t-\tau)=ae^{-at}\sum_{n=0}^{\infty}(2n+1)P_n^*(\tau)P_n^*(t) \tag{1-28} \]

or, taking into account that \(\delta(t-\tau)=\delta(\tau-t)\), the series may be written in the form

\[ \delta(t-\tau)=ae^{-a\tau}\sum_{n=0}^{\infty}(2n+1)P_n^*(\tau)P_n^*(t). \tag{1-29} \]

Expansions (1-28) and (1-29) must be understood in the sense of weak convergence [2]. For \(\tau=0\) we obtain, taking (1-5) into account,

\[ \delta(t)=ae^{-at}\sum_{n=0}^{\infty}(2n+1)(-1)^nP_n^*(t) =a\sum_{n=0}^{\infty}(2n+1)(-1)^nP_n^*(t) \tag{1-30} \]

or, more precisely,

\[ \delta(t)=ae^{-a|t|}\sum_{n=0}^{\infty}(2n+1)(-1)^nP_n^*(|t|), \tag{1-31} \]

since \(\delta(t)\) is an even function. Note that the same result can be obtained directly by determining the coefficients from the general formula (1-9).

Let us find the expansion of the unit function with a shifted argument, i.e., of the function

\[ f(t)=1(t-\tau)= \begin{cases} 0, & 0\le t<\tau,\\ 1, & \tau\le t<\infty. \end{cases} \]

For the coefficients we have

\[ C_n=a(2n+1)\int_0^\infty 1(t-\tau)e^{-at}P_n^*(t)\,dt =a(2n+1)\int_\tau^\infty e^{-at}P_n^*(t)\,dt. \]

Taking (1-4) into account, we obtain

\[ c_0=e^{-a\tau},\qquad c_n=-\frac{1}{2}\left[P_{n+1}^*(\tau)-P_{n-1}^*(\tau)\right]\quad (n=1,2\ldots). \]

The desired expansion has the form

\[ 1(t-\tau)=e^{-a\tau}-\frac{1}{2}\sum_{n=1}^{\infty} \left[P_{n+1}^*(\tau)-P_{n-1}^*(\tau)\right]P_n^*(\tau). \tag{1-9} \]

The series converges in the mean.

Up to now we have not at all touched on the question of the value of the parameter “\(a\).” It may be asserted that the expansion of an arbitrary function in “\(e\)”-Legendre polynomials will converge independently of the value of the parameter “\(a\),” provided condition (1-8) is satisfied. However, if one restricts oneself to a finite number of terms, then the accuracy of the approximation will already depend substantially on the value of the parameter “\(a\),” and the more so, the smaller the segment of the series taken. The parameter “\(a\)” can be chosen from the condition of minimizing the mean-square error of approximation by a segment of the series. In practice, however, such a path leads to very cumbersome expressions and proves unsuitable. The weight factor \(e^{-at}\) ensures maximum approximation accuracy on the initial interval; therefore, if the parameter “\(a\)” is found from the condition that a segment of the series be exactly equal to the value \(f(t)\) for \(t=\infty\), then, as calculations show, this value of the parameter “\(a\)” will be close to optimal.

Thus, the problem reduces to solving the equation

\[ \sum_{n=0}^{m} c_n P_n^*(\infty)=\sum_{n=0}^{m}c_n=f(\infty)\ne\infty \]

(\(c_n\) depends on “\(a\)”). This equation can be solved at least approximately.

2. “\(e\)”-Chebyshev polynomials of the first kind

The exponential Chebyshev polynomials of the first kind are a particular solution of the differential equation

\[ \frac{d}{dt}\left[ \sqrt{e^{at}(1-e^{-at})}\,\frac{dT_n^*(t)}{dt} \right] + \frac{e^{-\frac{at}{2}}\,n^2a^2T_n^*(t)} {\sqrt{1-e^{-at}}} =0. \tag{2-1} \]

Just as in the case of the “\(e\)”-Legendre polynomials, these polynomials can be obtained from the ordinary Chebyshev polynomials of the first kind by the substitution *) \(x\to 2e^{-at}-1\). All formulas derived for the ordinary polynomials remain valid. In particular,

*) The substitution \(x\to 1-2e^{-at}\) is possible. In this case \(T_n^*(t)\) must be multiplied by \((-1)^n\).

\[ T_n^*(t)=\cos n\arccos(2e^{-at}-1), \tag{2-2} \]

\[ T_{n+1}^*(t)=2(2e^{-at}-1)T_n^*(t)-T_{n-1}^*(t). \tag{2-3} \]

Introduce the substitution

\[ \alpha=\arccos(2e^{-at}-1), \tag{2-4} \]

i.e.,

\[ e^{-at}=\frac{1+\cos\alpha}{2}=\cos^2\frac{\alpha}{2}. \tag{2-4} \]

Then, as follows from (2-2), the sequence of “\(e\)-polynomials” \(T_n^*(t)\) \((n=0,1,2,\ldots)\) is transformed into the sequence of cosines \(\cos n\alpha\). It also follows from (2-2) that

\[ T_n^*(0)=1;\quad T_n^*(\infty)=(-1)^n;\quad |T_n^*(t)|\leqslant 1. \tag{2-5} \]

An explicit expression for the polynomials \(T_n^*(t)\) can be obtained by expanding the corresponding hypergeometric series

\[ T_n^*(t)=(-1)^n F\left(-n,n;\frac{1}{2};e^{-at}\right)=(-1)^n n\times \]

\[ \times\sum_{k=0}^{n}\frac{(-1)^k2^k(n+k-1)!\,e^{-kat}}{(n-k)!(2k-1)!!\,k!}. \tag{2-6} \]

For the first several values of \(n\) we shall have \(\left[T_0^*(t)=\frac{1}{2}\right]\).

\[ \left. \begin{aligned} T_1^*(t)&=-1+2e^{-at},\\ T_2^*(t)&=1-8e^{-at}+8e^{-2at},\\ T_3^*(t)&=-1+18e^{-at}-48e^{-2at}+32e^{-3at},\\ T_4^*(t)&=1-32e^{-at}+160e^{-2at}-252e^{-3at}+128e^{-4at},\\ T_5^*(t)&=-1+50e^{-at}-400e^{-2at}+1120e^{-3at}-1280e^{-4at}\\ &\quad+512e^{-5at} \end{aligned} \right\} \tag{2-7} \]

Note that the coefficients of the polynomials \(T_n^*(t)\) coincide with the coefficients of the shifted Chebyshev polynomials of the first kind, defined on the interval \((0,1)\) [6,9].

As already noted, the system of polynomials \(T_n^*(t)\) is complete in the space \(L^2_{W(t)}\), where the weight function \(W(t)\) has the form

\[ W(t)=e^{-\frac{at}{2}}(1-e^{-at})^{-\frac{1}{2}}. \tag{2-8} \]

From the general theory of orthogonal expansions it follows [8] that if \(f(t)\) satisfies the condition

\[ \int_{0}^{\infty} W(t)|f(t)|^2\,dt<\infty, \tag{2-9} \]

then the series

\[ f(t)=\sum_{n=0}^{\infty} A_nT_n^*(t) \tag{2-10} \]

will converge to \(f(t)\) at every interior point at which \(f(t)\) is continuous, provided the coefficients \(A_n\) are determined by the formula

\[ A_n=\frac{2a}{\pi}\int_0^\infty \frac{e^{-\frac{at}{2}} f(t)}{\sqrt{1-e^{-at}}}\,T_n^*(t)\,dt. \tag{2-11} \]

If we take into account that \(T_0^*(t)=\dfrac{1}{2}\), then the series (2-10) can be written in the form

\[ f(t)=\frac{A_0}{2}+\sum_{n=1}^{\infty} A_n T_n^*(t). \tag{2-12} \]

If, however, we carry out the substitution (2-4), we obtain

\[ f(t)\to f^*\left(-\frac{1}{a}\ln \cos^2 \frac{\alpha}{2}\right) = \frac{A_0}{2}+\sum_{n=1}^{\infty} A_n \cos n\alpha, \tag{2-13} \]

\[ A_n=\frac{2}{\pi}\int_0^\pi f^*\left(-\frac{1}{a}\ln \cos^2 \frac{\alpha}{2}\right)\cos n\alpha\,d\alpha. \tag{2-14} \]

Thus, we have the expansion of a certain periodic function
\(f^*\left(-\dfrac{1}{a}\ln \cos^2 \dfrac{\alpha}{2}\right)\)
in an ordinary Fourier series, which on the interval \((0,\pi)\) coincides with our function
\(f\left(-\dfrac{1}{a}\ln \cos^2 \dfrac{\alpha}{2}\right)\);
moreover, its analytic continuation into the negative range of values of \(\alpha\) is performed so that
\(f^*\left(-\dfrac{1}{a}\ln \cos^2 \dfrac{\alpha}{2}\right)\)
becomes an even function. This circumstance makes it possible to obtain at once the expansion in “\(e\)”-Chebyshev polynomials of the first kind from the corresponding trigonometric Fourier series, and also to extend various transformations and identities known from the theory of Fourier series to these series. In particular, we note Parseval’s equality

\[ \int_0^\infty \frac{e^{-\frac{at}{2}}[f(t)]^2}{\sqrt{1-e^{-at}}}\,dt = \frac{\pi}{2a} \left[ \frac{A_0^2}{2}+\sum_{n=1}^{\infty}A_n^2 \right], \tag{2-15} \]

since the square of the norm of \(T_n^*(t)\) in the space under consideration is equal to

\[ \int_0^\infty \frac{e^{-\frac{at}{2}}[T_n^*(t)]^2}{\sqrt{1-e^{-at}}}\,dt = \frac{\pi}{2a}. \tag{2-16} \]

We also note the rule for multiplying the series (2-12) by the polynomial \(T_m^*(t)\). If the expansion (2-12) holds, then

\[ f(t)\cdot T_m^*(t) = \frac{A_m}{2} + \frac{1}{2}\sum_{k=1}^{\infty} \left(A_{k+m}+A_{|k-m|}\right)T_k^*(t). \tag{2-17} \]

The formula follows from the rule for multiplying the series (2-13) by \(\cos m\alpha\) [5].

Let us consider the expansions of several functions in series of “\(e\)”-Chebyshev polynomials of the first kind. We begin with the expansion of the exponential function with a complex parameter. We have

\[ e^{-pt}=\frac{A_0(p)}{2}+\sum_{n=1}^{\infty} A_n(p)T_n^*(t). \tag{2-18} \]

The series will converge uniformly for any real value of “\(a\),” if \(\operatorname{Re} p > -\dfrac{a}{2}\). According to (2-11) and (2-14), one may write

\[ A_n(p)=\frac{2a}{\pi}\int_0^\infty \frac{e^{-pt}e^{-\frac{at}{2}}T_n^*(t)} {\sqrt{1-e^{-at}}}\,dt = \frac{2}{\pi}\int_0^\pi \left(\cos\frac{\alpha}{2}\right)^{\frac{2p}{a}}\cos n\alpha\,d\alpha = \]

\[ =\frac{4}{\pi}\int_0^{\frac{\pi}{2}}(\cos x)^{\frac{2p}{a}}\cos 2n\,dx. \tag{2-19} \]

The last integral is tabulated [4]:

\[ A_n(p)= \frac{4} {2^{2\left(\frac{p}{a}+1\right)} \left(\frac{p}{a}+\frac{1}{2}\right) \mathrm{B}\left(\frac{p}{a}+n+1,\frac{p}{a}+1-n\right)}. \tag{2-20} \]

The beta function has the representation

\[ \mathrm{B}\left(\frac{p}{a}+n+1,\frac{p}{a}+1-n\right) = \]

\[ = \frac{ \left(\frac{p}{a}+n\right)\left(\frac{p}{a}-n\right) \Gamma\left(\frac{p}{a}+n\right)\Gamma\left(\frac{p}{a}-n\right) } {\Gamma\left(2\frac{p}{a}+2\right)}. \]

After a number of transformations we shall have

\[ A_n(p)= \frac{ 2\Gamma\left(\frac{p}{a}+\frac{1}{2}\right) } { \sqrt{\pi}\,\Gamma\left(\frac{p}{a}\right) \left(\frac{p}{a}+n\right) } \prod_{\kappa=1}^{n-1} \frac{p-\kappa a}{p+\kappa a}, \tag{2-21} \]

where for \(n=0,1\)

\[ \prod_{\kappa=1}^{n-1}\frac{p-\kappa a}{p+\kappa a}=1. \]

The desired expansion takes the form

\[ e^{-pt}= \frac{ \Gamma\left(\frac{p}{a}+\frac{1}{2}\right) } { \sqrt{\pi}\,\Gamma\left(\frac{p}{a}\right) } \left[ \frac{a}{p} + 2\sum_{n=1}^{\infty} \frac{1}{\frac{p}{a}+n} \sum_{\kappa=1}^{n-1} \frac{p-\kappa a}{p+\kappa a}\, T_n^*(t) \right]. \tag{2-22} \]

From (2-19) it follows that

\[ A_n(p)\doteq \frac{2a}{\pi}\, \frac{e^{-\frac{at}{2}}}{\sqrt{1-e^{-at}}}\, T_n^*(t) = \frac{2a}{\pi}W(t)T_n^*(t), \tag{2-23} \]

i.e., \(A_n(p)\) is the Laplace transform of the function \(\dfrac{2a}{\pi}W(t)T_n^*(t)\). Accordingly,

\[ A_0(p)\doteq \frac{2a}{\pi}W(t). \tag{2-24} \]

Let \(\tau\) be a fixed value of \(t\); then

\[ e^{-p\tau}=\frac{A_0(p)}{2}+\sum_{n=1}^{\infty} A_n(p)T_n^*(t). \tag{2-25} \]

Taking into account that \(e^{-p\tau}\doteq \delta(t-\tau)\), and also (2-24) and (2-25), we obtain an expansion of the delta function in a series in the “\(e^t\)-polynomials” \(T_n^*(t)\).

\[ \delta(t-\tau)=\frac{a}{\pi}W(t)\left[1+2\sum_{n=1}^{\infty}T_n^*(\tau)T_n^*(t)\right], \tag{2-26} \]

or, in another form, if we bear in mind that \(\delta(t-\tau)=\delta(\tau-t)\),

\[ \delta(t-\tau)=\frac{a}{\pi}W(\tau)\left[1+2\sum_{n=1}^{\infty}T_n^*(\tau)\cdot T_n^*(t)\right]. \tag{2-27} \]

Let us note the special case when \(p=ma\) \((m=1,2\ldots)\). Expansion (2-22) in this case automatically terminates at \(n=m\) and takes the form

\[ e^{-mat}=\frac{\Gamma\left(m+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(m)} \left[ \frac{1}{m} +2\sum_{n=1}^{m}\frac{1}{m+n} \prod_{k=1}^{n-1}\frac{m-k}{m+k}\,T_n^*(T) \right]. \tag{2-28} \]

This formula gives a representation of the powers \((e^{-at})^m\) in the form of a linear combination of a finite number of Chebyshev polynomials \(T_n^*(t)\).

For the first few values of \(m\) we shall have:

\[ \left. \begin{aligned} e^{-at} &= \frac{1}{2}+\frac{1}{2}T_1^*(t),\\ e^{-2at} &= \frac{3}{8}+\frac{1}{2}T_1^*(t)+\frac{1}{8}T_2^*(t),\\ e^{-3at} &= \frac{5}{16}+\frac{15}{32}T_1^*(t)+\frac{3}{16}T_2^*(t)+\frac{1}{32}T_3^*(t),\\ e^{-4at} &= \frac{25}{128}+\frac{7}{16}T_1^*(t)+\frac{7}{32}T_2^*(t)+\frac{1}{16}T_3^*(t)+\frac{1}{128}T_4^*(t). \end{aligned} \right\} \tag{2-29} \]

From expansion (2-22), using formula (1-21), one can obtain an expansion for \(t\); however, it is simpler to use the trigonometric identity [5]

\[ -\ln\left(2\cos\frac{\alpha}{2}\right) = \sum_{k=1}^{\infty}\frac{(-1)^k}{k}\cos k\alpha. \]

If we carry out the substitution (2-4), then we obtain

\[ at=2\ln 2+2\sum_{k=1}^{\infty}\frac{(-1)^k}{k}\,T_k^*(t), \qquad 0\leq t<\infty. \tag{2-30} \]

Formula (2-17) gives

\[ at\cdot T_1^*(t) = -1+\left(\frac{1}{2}+2\ln 2\right)T_1^*(t) -2\sum_{k=2}^{\infty}\frac{(-1)^k k}{k^2-1}T_k^*(t). \tag{2-31} \]

Bearing in mind that

\[ ate^{-at}=\frac{1}{2}\left[at+at\,T_1^*(t)\right], \]

we easily find the expansion for \(ate^{-at}\)

\[ ate^{-at}=\left(\ln 2-\frac{1}{2}\right)+\left(\ln 2-\frac{3}{4}\right)T_1^*(t)-\sum_{\kappa=2}^{\infty}\frac{(-1)^\kappa}{\kappa(\kappa^2-1)}T_\kappa^*(t). \tag{2-32} \]

Let us also note some expansions that follow directly from the corresponding trigonometric identities. In particular, we may write the following representation of integral positive powers of the “\(e\)”-polynomial \(T_m^*(t)\):

\[ \left[T_m^*(t)\right]^{2n} = \frac{1}{2^{2n}} \left[ \sum_{k=0}^{n-1}2\binom{2n}{k}T_{2m(n-k)}^*(t)+\binom{2n}{n} \right], \]

\[ \left[T_m^*(t)\right]^{2n-1} = \frac{1}{2^{2n-2}} \sum_{k=0}^{n-1}\binom{2n-1}{k}T_{m(2n-2k-1)}^*(t). \tag{2-33} \]

The formulas are obtained from the expansions of \((\cos m\alpha)^{2n}\) and \((\cos m\alpha)^{2n-1}\) into Fourier series by passing to the variable \(t\) according to formula (2-4) [4]. The symbol \(\binom{2n}{k}\) denotes the binomial coefficient.

The following representation of the ordinary Legendre polynomials is known [4]:

\[ P_n(\cos\alpha) = \frac{(2n-1)!!}{2^{n-1}n!} \left[ \cos n\alpha + \frac{1}{1}\frac{n}{2n-1}\cos(n-2)\alpha + \right. \]

\[ \left. + \frac{1\cdot 3}{1\cdot 2} \frac{n(n-1)}{(2n-1)(2n-3)} \cos(n-4)\alpha+\cdots \right] = \]

\[ = \frac{(2n-1)!!}{2^{n-1}n!} \sum_{m=0}^{\left[\frac{n}{2}\right]} \frac{(2m-1)!!}{m!} \prod_{\kappa=0}^{m-1} \frac{n-\kappa}{2n-(2\kappa+1)} \cos(n-2m)\alpha. \]

Making the substitution (2-4), we obtain

\[ P_n^*(t) = (-1)^n \frac{(2n-1)!!}{2^{n-1}n!} \sum_{m=0}^{\left[\frac{n}{2}\right]} \frac{(2m-1)!!}{m!} \prod_{\kappa=0}^{m-1} \frac{n-\kappa}{2n-(2\kappa+1)} T_{n-2m}^*(t), \tag{2-34} \]

where \(\prod_{\kappa=0}^{m-1}\equiv 1\) for \(m=0\), and \(\left[\frac{n}{2}\right]\) denotes the integer part. Formula (2-34) gives a representation of the “\(e\)”-Legendre polynomial in the form of a linear combination of the “\(e\)”-Chebyshev polynomials of the first kind.

3. “\(e\)”-Chebyshev polynomials of the second kind

These polynomials may be defined by the relation

\[ U_n^*(t)=\frac{1}{2(n+1)}\frac{d}{d(e^{-at})}T_{n+1}^*(t) = \frac{\sin(n+1)\arccos(2e^{-at}-1)} {2e^{-\frac{at}{2}}\sqrt{1-e^{-at}}}. \tag{3-1} \]

The substitution (2-4) gives the expression

\[ U_n^*(t)\to \frac{\sin(n+1)\alpha}{\sin\alpha}. \tag{3-2} \]

The polynomials \(U_n^*(t)\) satisfy the differential equation

\[ \frac{d}{dt}\left[e^{-\frac{at}{2}}\left(1-e^{-at}\right)^{\frac{3}{2}} \frac{dU_n^*(t)}{dt}\right] +n(n+2)a^2 e^{-\frac{3}{2}at}\left(1-e^{-at}\right)^{\frac{1}{2}}U_n^*(t)=0. \tag{3-3} \]

The basic recurrence formula has the form

\[ U_{n+1}^*(t)=2\left(2e^{-at}-1\right)U_n^*(t)-U_{n-1}^*(t). \tag{3-4} \]

Earlier we defined \(U_n^*(t)\) as a special case of the Jacobi “\(e\)-polynomials”; however, in order to be consistent with the preceding formulas, it is necessary to put

\[ U_n^*(t)=(-1)^n(n+1)F\left(-n,n+2;\frac{3}{2};e^{-at}\right) \tag{3-5} \]

or, in expanded form,

\[ U_n^*(t)=(-1)^n\sum_{\kappa=0}^{n} \frac{(-1)^\kappa 2^\kappa(n+\kappa+1)!\,e^{-\kappa at}} {(2\kappa+1)(n-\kappa)!\,\kappa!\,(2\kappa-1)!!}, \tag{3-5'} \]

whence it follows that

\[ U_n^*(0)=n+1 \quad \text{and} \quad U_n^*(\infty)=(-1)^n(n+1). \tag{3-6} \]

The polynomials \(U_n^*(t)\) can be expressed in terms of the “\(e\)-polynomials” \(T_n^*(t)\). The identities [4]

\[ \frac{\sin 2n\alpha}{\sin\alpha} = 2\sum_{k=1}^{n}\cos(2k-1)\alpha, \]

\[ \frac{\sin(2n+1)\alpha}{\sin\alpha} = 2\sum_{k=0}^{n}\cos 2k\alpha - 1 \]

hold. Taking into account (2-4), and also the fact that \(T_0^*(t)=\frac{1}{2}\), we obtain

\[ \left. \begin{aligned} U_{2n-1}^*(t)&=2\sum_{k=1}^{n}T_{2k-1}^*(t) \qquad (n=1,2,\ldots),\\[4pt] U_{2n}^*(t)&=2\sum_{k=0}^{n}T_{2k}^*(t) \qquad (n=0,1,\ldots). \end{aligned} \right\} \tag{3-7} \]

Let us also note the following relations, which follow from the corresponding trigonometric identities:

\[ \left. \begin{aligned} U_n^*(t)\cdot T_m^*(t) &= \frac{1}{2}\left[U_{n+m}^*(t)+U_{n-m}^*(t)\right],\\[4pt] T_{n+1}^*(t)\cdot U_{m-1}^*(t) &= \frac{1}{2}\left[U_{n+m}^*(t)-U_{n-m}^*(t)\right]. \end{aligned} \right\} \tag{3-8} \]

Putting \(m=1\) in the last formula, we obtain

\[ T_{n+1}^*(t)=\frac{1}{2}\left[U_{n+1}^*(t)-U_{n-1}^*(t)\right] \qquad (n=1,2\ldots) \]

or

\[ T_n^*(t)=\frac{1}{2}\left[U_n^*(t)-U_{n-2}^*(t)\right] \qquad (n=2,3,\ldots). \tag{3-9} \]

As already noted, the “\(e\)-polynomials” \(U_n^*(t)\) form a complete system, orthogonal on the interval \((0,\infty)\) with weight \(e^{-\frac{3}{2}at}\sqrt{1-e^{-at}}\), i.e.

\[ \int_0^\infty e^{-\frac{3}{2}at}\sqrt{1-e^{-at}}\,U_n^*(t)\,U_m^*(t)\,dt = \begin{cases} 0, & m\ne n,\\[4pt] \dfrac{\pi}{8a}, & m=n. \end{cases} \tag{3-10} \]

Any piecewise-continuous function satisfying the condition

\[ \int_0^\infty e^{-\frac{3}{2}at}\sqrt{1-e^{-at}}\,|f(t)|^2\,dt<\infty \]

can be expanded in a series in the polynomials \(U_n^*(t)\)

\[ f(t)=\sum_{n=0}^{\infty} D_n U_n^*(t), \tag{3-11} \]

where

\[ D_n=\frac{8a}{\pi}\int_0^\infty e^{-\frac{3}{2}at}\sqrt{1-e^{-at}}\,f(t)\,U_n^*(t)\,dt \tag{3-12} \]

or, in another form, if the substitution (2-4) is made,

\[ D_n=\frac{2}{\pi}\int_0^\pi f\!\left(-\frac{1}{a}\ln \cos^2\frac{\alpha}{2}\right) \sin\alpha\cdot \sin(n+1)\alpha\,d\alpha . \tag{3-13} \]

If we take into account the identity

\[ \sin(n+1)\alpha\cdot \sin\alpha = \frac{1}{2}\cos n\alpha-\frac{1}{2}\cos(n+2)\alpha \]

and use (2-14), we obtain

\[ D_n=\frac{1}{2}\left(A_n-A_{n-2}\right)\quad (n=0,1,2,\ldots). \tag{3-14} \]

Thus, the coefficients in the expansion of \(f(t)\) in the “e”-polynomials \(U_n^*(t)\) are related to the coefficients in the expansion of the same function in the “e”-Chebyshev polynomials of the first kind by the elementary formula (3-14).

4. Exponential Chebyshev Functions of the Third Kind

If, in equation (2-1), the independent variable is transformed according to (2-4), it takes the form

\[ \frac{d^2y}{d\alpha^2}+n^2y=0,\qquad 0\le \alpha\le \pi. \]

Two linearly independent solutions of this equation are

\[ y_{\mathrm{I}}=\cos n\alpha,\qquad y_{\mathrm{II}}=\sin n\alpha. \]

If we return to the former variable, the first particular solution is transformed into the sequence of “e”-Chebyshev polynomials of the first kind, \(T_n^*(t)\), while the second is transformed into the sequence of functions of the form

\[ S_n(t)=2e^{-\frac{at}{2}}\sqrt{1-e^{-at}}\,U_{n-1}^*(t), \tag{4-1} \]

which we shall call exponential Chebyshev functions of the third kind. These functions will no longer be polynomials, as was the case for \(T_n^*(t)\) and \(U_n^*(t)\). Their principal property consists in the fact that, being eigenfunctions of the differential opera-

the Sturm–Liouville operator, form on the interval \((0,\infty)\) a complete orthogonal system with weight

\[ W(t)=e^{-\frac{at}{2}}\left(1-e^{-at}\right)^{-\frac12}. \]

Indeed, taking into account (4-1) and (3-10), we shall have

\[ \int_{0}^{\infty} \frac{e^{-\frac{at}{2}} S_n(t)\cdot S_m(t)} {\sqrt{1-e^{-at}}}\,dt = 4\int_{0}^{\infty} e^{-\frac32 at}\sqrt{1-e^{-at}}\, U_{n-1}^{*}(t)U_{m-1}^{*}(t)\,dt = \begin{cases} 0, & m\ne n,\\[4pt] \dfrac{\pi}{2a}, & m=n. \end{cases} \]

The basic functional relations for \(S_n(t)\), just as for \(T_n^{*}(t)\), are obtained directly from the well-known trigonometric identities. In particular, we have

\[ \begin{aligned} S_{n\pm m}(t)&=S_n(t)\cdot T_m^{*}(t)\pm T_n^{*}(t)\cdot S_m(t),\\ T_{n\pm m}^{*}(t)&=T_n^{*}(t)\cdot T_m^{*}(t)+S_n(t)\cdot S_m(t),\\ S_n(t)\cdot S_m(t)&=\frac12\left[T_{n-m}^{*}(t)-T_{n+m}^{*}(t)\right],\\ S_n(t)\cdot T_m^{*}(t)&=\frac12\left[S_{n-m}(t)-S_{n+m}(t)\right]. \end{aligned} \tag{4-2} \]

Putting \(m=1\) in the last relation and taking into account that

\[ T_1^{*}(t)=2e^{-at}-1, \]

we obtain the basic recurrence formula for the functions under consideration

\[ 2(2e^{-at}-1)S_n(t)=S_{n-1}(t)+S_{n+1}(t). \tag{4-3} \]

From (4-1) it follows that: \(S_n(0)=S_n(\infty)=0\). The following identities hold [5]:

\[ \sin(2n+1)\alpha= \frac{2}{\pi(2n+1)} - \frac{4(2n+1)}{\pi} \sum_{m=1}^{\infty} \frac{\cos 2m\alpha}{(2m)^2-(2n+1)^2}, \quad (n=0,1,2\ldots), \]

\[ \sin 2n\alpha = -\frac{8n}{\pi} \sum_{m=1}^{\infty} \frac{\cos(2m-1)\alpha}{(2m-1)^2-(2n)^2} \quad (n=1,2,\ldots). \]

They are expansions of sinusoidal “packets,” given on the interval \((0,\pi)\), in a cosine series. Transformation (2-4) gives

\[ \left. \begin{aligned} S_{2n+1}(t) &= \frac{2}{\pi(2n+1)} - \frac{4(2n+1)}{\pi} \sum_{m=1}^{\infty} \frac{T_{2m}^{*}(t)}{(2m)^2-(2n+1)^2}, \quad (n=0,1,2\ldots),\\[6pt] S_{2n}(t) &= -\frac{8n}{\pi} \sum_{m=1}^{\infty} \frac{T_{2m-1}^{*}(t)}{(2m-1)^2-(2n)^2} \quad (n=1,2,3\ldots), \end{aligned} \right\} \tag{4-4} \]

i.e., the functions \(S_n(t)\) \((n=1,2,3\ldots)\) are expanded in infinite series in the \(e\)-polynomials of Chebyshev of the first kind.

Let us consider the question of expanding an arbitrary function in a series in the functions \(S_n(t)\).

Condition (2-9) turns out to be sufficient for the convergence of the expansion

\[ f(t)=\sum_{k=1}^{\infty} B_k S_k(t) \tag{4-5} \]

at every point that is a point of continuity of \(f(t)\), if the coefficients \(B_k\) are defined by the formula

\[ B_k=\frac{2a}{\pi}\int\limits_0^\infty \frac{e^{-\frac{at}{2}}f(t)S_k(t)}{\sqrt{1-e^{-at}}}\,dt = \frac{2}{\pi}\int\limits_0^\pi f\left(-\frac{1}{a}\ln\cos^2\frac{\alpha}{2}\right)\sin k\alpha\,d\alpha . \tag{4-6} \]

At discontinuities of the first kind, the series (4-5) converges to the value
\(\frac{1}{2}[f(t_i+0)-f(t_i-0)]\). If the change of variable according to (2-4) is made, then (4-5) is transformed into the form

\[ f\left(-\frac{1}{a}\ln\cos^2\frac{\alpha}{2}\right) = \sum\limits_{k=1}^{\infty}B_k\sin k\alpha, \qquad 0\leq \alpha\leq \pi . \tag{4-7} \]

The series (4-7) may be regarded as the expansion of a certain periodic function
\(f^*\left(-\frac{1}{a}\ln\cos^2\frac{\alpha}{2}\right)\), which on the interval \((0,\pi)\) coincides with the given function and which we have constructed in such a way that it becomes odd.

As an example, let us consider the expansion of the unit step function \(1(t-\tau)\). The expansion coefficients will be equal to

\[ B_k=\frac{2a}{\pi}\int\limits_{\tau}^{\infty} \frac{e^{-\frac{at}{2}}S_k(t)}{\sqrt{1-e^{-at}}}\,dt = \frac{2}{\pi}\int\limits_{\alpha_\tau}^{\pi}\sin k\alpha\,d\alpha = \frac{2\left[T_k^*(\tau)-(-1)^k\right]}{\pi k}, \tag{4-8} \]

since \(\cos k\alpha_\tau=T_k^*(\tau)\). The expansion has the form

\[ 1(t-\tau)=\frac{2}{\pi}\sum\limits_{k=1}^{\infty} \frac{T_k^*(\tau)-(-1)^k}{k}\,S_k(t). \tag{4-9} \]

In the special case when \(\tau=0\), after transformations we obtain

\[ 1(t)=\frac{4}{\pi}\sum\limits_{k=1}^{\infty} \frac{1}{2k-1}\,S_{2k-1}(t). \tag{4-10} \]

Numerous expansions can be obtained from the corresponding trigonometric series and identities by a change of variable.

Conclusion

The properties of the orthogonal functions considered and the various examples, which are of independent interest and illustrate the technique of expanding diverse functions into generalized Fourier series, show that the proposed analytical apparatus can be effectively used not only for purposes of function approximation, but also for solving other problems of a theoretical and applied nature.

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