# Euler transformation

The Euler transformation of series. Given a series

$$\tag{1 } \sum _ { n= } 0 ^ \infty (- 1) ^ {n} a _ {n} ,$$

the series

$$\tag{2 } \sum _ { n= } 0 ^ \infty \frac{\Delta ^ {n} a _ {0} }{2 ^ {n+} 1 }$$

is said to be obtained from (1) by means of the Euler transformation. Here

$$\Delta ^ {n} a _ {0} = \sum _ { k= } 0 ^ { n } (- 1) ^ {k} \left ( \begin{array}{c} n \\ k \end{array} \right ) a _ {k} .$$

If the series (1) converges, then so does (2), and to the same sum as (1). If the series (2) converges (in this case (1) may diverge), then the series (1) is called Euler summable.

If (1) converges, if $a _ {n} > 0$, if the sequences

$$\Delta ^ {k} a _ {n} = \sum _ { l= } 0 ^ { k } ( - 1) ^ {l} \left ( \begin{array}{c} k \\ l \end{array} \right ) a _ {n+} l ,\ \ k = 0 , 1 \dots$$

are monotone, and if

$$\frac{a _ {n+} 1 }{a _ {n} } \geq q > \frac{1}{2} ,$$

then the series (2) converges more rapidly than (1) (see Convergence, types of).

L.D. Kudryavtsev

Euler's transformation is the integral transformation

$$\tag{1 } w ( z) = \int\limits _ { C } ( z - t ) ^ \alpha v ( t) dt ,$$

where $C$ is a contour in the complex $t$- plane. It was proposed by L. Euler (1769).

The Euler transformation is applied to linear ordinary differential equations of the form

$$\tag{2 } L w = \sum _ { j= } 0 ^ { n } (- 1) ^ {j} w ^ {(} j) \sum _ { k= } 0 ^ { n- } j \left ( \begin{array}{c} n + \beta - j - 1 \\ n - k - j \end{array} \right ) Q _ {j} ^ {(} n- k- j) ( z) = 0 ,$$

where $Q _ {j} ( z)$ is a polynomial of degree $\leq n - j$ and $\beta$ is a constant. Any linear equation of the form

$$P _ {n} ( z) w ^ {(} n) + P _ {n-} 1 ( z) w ^ {(} n- 1) + \dots + P _ {0} ( z) w = 0 ,$$

where the $P _ {j} ( z)$ are polynomials of degree $\leq j$ and the degree of $P _ {n} ( z)$ is $n$, can be written in the form (2). The equation

$$M v \equiv \sum _ { j= } 0 ^ { n } (- 1) ^ {j} ( Q _ {n-} j ( z) v ) ^ {(} j) = 0$$

is called the Euler transform of (2). If $w ( z)$ is defined by (1) and $\alpha = \beta + n - 1$, then

$$L w = \int\limits _ { C } ( z - t ) ^ \alpha M ( v) dt ,$$

provided that the integrated term arising from integration by parts vanishes. From this it follows that if $M ( v) = 0$, then $w ( z)$ is a solution of (2).

The Euler transformation makes it possible to reduce the order of (2) if $Q _ {n-} j ( z) \equiv 0$ for $j > q$, $q < n$. For $q = 0$ and $q= 1$ equation (2) can be integrated (see Pochhammer equation).

## Contents

How to Cite This Entry:
Euler transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_transformation&oldid=46861
This article was adapted from an original article by L.D. Kudryavtsev, M.V. Fedoryuk, Yu.A. Brychkov, A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article