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Tytuł książki
Tytuł rozdziału
A remark on solutions of the laguerre differential equation
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Treść / Zawartość
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EN
Abstract: The Kummer solution $x_{a}$ of the Laguerre differential equation
t(d²x/dt²) + (1-t)(dx/dt) - ax = 0, a ∈ ℂ,
can be represented by means of the power series
$x_{a} = 1 + (at/1!) + (a(a+1)/2!)·(t²/2!) + ... + ((a(a+1)...[(a+n)-1])/n!)·(t^{n}/n!) + ... $
Let $x_{a,γ}(t) = e^{γt} x_{a}(t)$. Applying Mikusiński's operational calculus the formulas
$d/dt [x_{a}*x_{b}](t) = x_{a+b}(t)$,
$(d/dt - γ)[x_{a,γ}*x_{b,γ}](t) = x_{a+b,γ}(t)$
will be obtained.
In particular, we will obtain relations between Laguerre polynomials $L_{n}$, $L_{m}$ and Laguerre functions $l_{m}$, $l_{n}$ of degree n and m which can be written in the following form:
$(d/dt)(L_{n}*L_{m})(t) = L_{n+m}(t)$,
$(d/dt + 1/2)(l_{n}*l_{m})(t) = l_{n+m}(t)$,
where the star * denotes convolution on the positive half line.
t(d²x/dt²) + (1-t)(dx/dt) - ax = 0, a ∈ ℂ,
can be represented by means of the power series
$x_{a} = 1 + (at/1!) + (a(a+1)/2!)·(t²/2!) + ... + ((a(a+1)...[(a+n)-1])/n!)·(t^{n}/n!) + ... $
Let $x_{a,γ}(t) = e^{γt} x_{a}(t)$. Applying Mikusiński's operational calculus the formulas
$d/dt [x_{a}*x_{b}](t) = x_{a+b}(t)$,
$(d/dt - γ)[x_{a,γ}*x_{b,γ}](t) = x_{a+b,γ}(t)$
will be obtained.
In particular, we will obtain relations between Laguerre polynomials $L_{n}$, $L_{m}$ and Laguerre functions $l_{m}$, $l_{n}$ of degree n and m which can be written in the following form:
$(d/dt)(L_{n}*L_{m})(t) = L_{n+m}(t)$,
$(d/dt + 1/2)(l_{n}*l_{m})(t) = l_{n+m}(t)$,
where the star * denotes convolution on the positive half line.
Słowa kluczowe
Kategorie tematyczne
Twórcy
autor
- Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
autor
- Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8, 40-013 Katowice, Poland
Strony
Bibliografia
[1] H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. I, Nauka, Moscow, 1973 (in Russian).
[2] V. A. Ditkin and A. P. Prudnikov, Operational Calculus, Moscow, 1975 (in Russian).
[3] H. J. Glaeske, On the Wiener-Laguerre transform of generalized functions, in: Generalized Functions and Convergence, World Scientific, 1990, 127-140.
[4] J. Mikusiński, Operational Calculus, Vol. I, PWN-Pergamon Press, Warszawa, 1983.
[5] J. Mikusiński and T. K. Boehme, Operational Calculus, Vol. II, PWN-Pergamon Press, Warszawa, 1987.
[6] N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, Technological Press of MIT and Wiley, New York, 1949.
[7] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
[8] K. Yosida, Operational Calculus. A Theory of Hyperfunctions, Springer, Berlin, 1984.
[2] V. A. Ditkin and A. P. Prudnikov, Operational Calculus, Moscow, 1975 (in Russian).
[3] H. J. Glaeske, On the Wiener-Laguerre transform of generalized functions, in: Generalized Functions and Convergence, World Scientific, 1990, 127-140.
[4] J. Mikusiński, Operational Calculus, Vol. I, PWN-Pergamon Press, Warszawa, 1983.
[5] J. Mikusiński and T. K. Boehme, Operational Calculus, Vol. II, PWN-Pergamon Press, Warszawa, 1987.
[6] N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, Technological Press of MIT and Wiley, New York, 1949.
[7] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
[8] K. Yosida, Operational Calculus. A Theory of Hyperfunctions, Springer, Berlin, 1984.
Kolekcja
DML-PL
Identyfikator YADDA
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