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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|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|Pochhammer equation]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Teubner  (1943)</TD></TR>
</table>

''M.V. Fedoryuk''

The Euler transform of the first kind is the integral transform

$$ 
I  ^  \mu  f ( x)  =  \mathfrak K _  \mu  \{ f ( t) ;  x \}  = \ 

\frac{1}{\Gamma ( \mu ) }
 \int\limits _ { 0 } ^ { x }  f ( t)
( x - t ) ^ {\mu - 1 }  dt ,
$$

where  $  \mu $
and  $  x $
are complex variables and the path of integration is the segment  $  t = x \tau $, 
$  0 < \tau < 1 $.

The Euler transform of the first kind is also called the fractional Riemann–Liouville integral of order  $  \mu $. 
(Sometimes the name of Riemann–Liouville integral is given to

$$ 

\frac{1}{\Gamma ( \mu ) }
 \int\limits _ { x } ^ { a }  f ( t)
( t - x ) ^ {\mu - 1 }  dt  =  \mathfrak K _  \mu  \{ f ( a - t ) ; \ 
a - x \} ,
$$

where  $  a $
is a complex number.)

If  $  f $
and  $  g $
satisfy certain conditions, then

$$ 
I  ^ {0} f ( x)  =  f ( x) ,
$$

$$ 
I  ^  \mu  [ \alpha f ( x) + \beta g ( x) ]  =  \alpha I  ^  \mu  f ( x) + \beta I  ^  \mu  g ( x) ,
$$

where  $  \alpha $
and  $  \beta $
are complex constants and

$$ 
I  ^  \mu  [ I  ^  \nu  f ( x) ]  =  I ^ {\mu + \nu } f ( x) ,
$$

$$ 
I  ^ {n} f ( x)  =  \int\limits _ { 0 } ^ { x }  dt _ {1} \dots
\int\limits _ { 0 } ^ { {t _ n-} 2 } d t _ {n-} 1 \int\limits _ { 0 } ^ { {t } _ {n-} 1 } f ( t _ {n} )  d t _ {n} ,
$$

$$ 
I  ^ {-} n f ( x)  =  
\frac{d  ^ {n} }{d x  ^ {n} }
 f ( x) ,\  n = 1 , 2 ,\dots .
$$

The Euler transform of the second kind is the integral transform

$$ 
K  ^  \mu  f ( x)  =  \mathfrak M _  \mu  \{ f ( t) ;  x \}  = \ 

\frac{1}{\Gamma ( \mu ) }
 \int\limits _ { x } ^  \infty   f ( t) ( t - x ) ^ {\mu - 1 }  d t ,
$$

where  $  \mu $
and  $  x $
are complex variables and the path of integration is the ray  $  t = x \tau $, 
$  \tau > 1 $, 
or  $  t = x + \tau $, 
$  \tau > 0 $. 
Under certain conditions,

$$ 
K  ^ {0} f ( x)  =  f ( x) ,
$$

$$ 
K  ^  \mu  [ \alpha f ( x) + \beta g ( x) ]  =  \alpha K  ^  \mu  f ( x) + \beta K  ^  \mu  ( x) ,
$$

where  $  \alpha $
and  $  \beta $
are complex constants and

$$ 
K  ^  \mu  [ K  ^  \nu  f ( x) ]  =  K ^ {\mu + \nu } f ( x) ,
$$

$$ 
K  ^ {n} f ( x)  =  \int\limits _ { x } ^  \infty   d t _ {1} \dots \int\limits _ {t _ {n-} 2 } ^  \infty  
d t _ {n-} 1 \int\limits _ {t _ {n-} 1 } ^  \infty   f ( t _ {n} )  d t _ {n} ,
$$

$$ 
K  ^ {-} n f ( x)  =  
\frac{d  ^ {n} }{d x  ^ {n} }
 f ( x) ,\  n = 1 , 2 ,\dots .
$$

The Euler transform of the second kind is sometimes called the fractional Weyl integral of order  $  \mu $.

The above transforms have also been introduced for generalized functions.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  Y.A. Brychkov,   A.P. Prudnikov,   "Integral transformations of generalized functions" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR>
</table>

''Yu.A. BrychkovA.P. Prudnikov''

====Comments====
See also [[Fractional integration and differentiation]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Erdélyi,   W. Magnus,   F. Oberhetinger,   F.G. Tricomi,   "Tables of integral transforms" , '''II''' , McGraw-Hill  (1954)  pp. Chapt. 13</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. McBride,   "Fractional calculus and integral transforms of generalized functions" , Pitman  (1979)</TD></TR>
</table>