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Register a new userThe Peano-Baker series
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Michael Baake
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This note reviews the Peano-Baker series and its use to solve the general linear system of ODEs. The account is elementary and self-contained, and is meant as a pedagogic introduction to this approach, which is well known but usually treated as a folklore result or as a purely formal tool. Here, a simple convergence result is given, and two examples illustrate that the series can be used explicitly as well.
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We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. For example, we construct several series whose sums remain unchanged when the nth term is multiplied by sin(n)/n. One series with this property is this classic series for pi/4: pi/4 = 1 - 1/3 + 1/5 ... = 1*(sin(1)/1) - (1/3)*(sin(3)/3) + (1/5)*(sin(5)/5).... Another example is sum (n = 1 to infinity) of (sin(n)/n) = sum (n = 1 to infinity) of (sin(n)/n)^2 = (pi - 1)/2. This material should be accessible to undergraduates. This paper also includes a Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9s is also a convergent series. We show how to compute sums of Irwins series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.007x10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol.
The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujans continued fraction formulae involving the quotient of the gamma functions.
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