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{{MSC|60E99}}

[[Category:Distribution theory]]

A series defined by the expression

$$ \tag{1 }
f _ {A} ( x)  = \ 
f ( x) +
\sum _ {k = 3 } ^ { n } 
a _ {k} f ^ { ( k) } ( x)
$$

or

$$ \tag{2 }
f _ {B} ( x)  = \ 
\psi ( x)
\sum _ {m = 0 } ^ { n } 
b _ {m} g _ {m} ( x),
$$

where  $  x $
is the normalized value of a random variable.

The series (1) is known as the Gram–Charlier series of type  $  A $; 
here

$$ 
f ( x)  = \ 
{
\frac{1}{\sqrt {2 \pi }}
 }
e ^ {- x  ^ {2} /2 } ,
$$

$  f ^ { ( k) } $
is the  $  k $-th derivative of  $  f $, 
which can be represented as

$$ 
f ^ { ( k) } ( x)  = \ 
(- 1)  ^ {k} H _ {k} ( x) f ( x),
$$

where  $  H _ {k} ( x) $
are the Chebyshev–Hermite polynomials. The derivatives  $  f ^ { ( k) } $
and the polynomials  $  H _ {k} $
are orthogonal, owing to which the coefficients  $  a _ {k} $
can be defined by the basic moments  $  r _ {k} $
of the given distribution series. If one restricts to the first few terms of the series (1), one obtains

$$ 
f _ {A} ( x)  = \ 
f ( x) + 
\frac{r _ {3} }{3!}

f ^ { ( 3) } ( x) +
$$

$$ 
+ 

\frac{r _ {4} - 3 }{4! }
 f ^ { ( 4) } ( x) - 
\frac{r _ {5} - 10r _ {3} }{5! }
 f ^ { ( 3) } ( x) + 
\frac{r _ {4} - 15r _ {4} + 30 }{6! }
 f ^ { ( 6) } ( x).
$$

The series (2) is known as a Gram–Charlier series of type  $  B $; 
here

$$ 
\psi ( x)  = \ 

\frac{\lambda  ^ {x} }{x!}

e ^ {- \lambda } ,\ \ 
x = 0, 1 \dots
$$

while  $  g _ {m} ( x) $
are polynomials analogous to the polynomials  $  H _ {k} ( x) $.

If one restricts to the first terms of the series (2), one obtains

$$ 
f _ {B} ( x)  = \ 

\frac{\lambda  ^ {x} }{x! }

e ^ {- \lambda } \left \{
1 + 
\frac{\mu _ {2} - \lambda }{\lambda  ^ {2} }

\left [ 
\frac{x  ^ {[2]} }{2 }
 -
\lambda x  ^ {[1]} +

\frac{\lambda  ^ {2} }{2 }
 \right ] \right . +
$$

$$ 
+ \left .

\frac{\mu _ {3} - 3 \mu _ {2} + 2 \lambda }{\lambda  ^ {3} }
 \left [ 
\frac{x  ^ {[3]} }{6 }
 -
{
\frac \lambda {2}
 } x  ^ {[2]} + 
\frac{\lambda  ^ {2} }{2 }
 x  ^ {[1]} - 
\frac{\lambda  ^ {3} }{6 }
 \right ]  \right \} .
$$

Here  $  \mu _ {i} $
are the central moments of the distribution, while  $  x  ^ {[i]} = x( x - 1) \dots ( x - i + 1) $.

Gram–Charlier series were obtained by J.P. Gram {{Cite|G}} and C.V.L. Charlier {{Cite|Ch}} in their study of functions of the form

$$ 
B _ {0} ( x)  = \ 
{
\frac{1}{2 \pi }
 }
\int\limits _ {- \pi } ^ { {+ }  \pi }
e ^ {- itx } \phi ( t)  dt.
$$

These are convenient for the interpolation between the values  $  B ( m) = ( n!/m! ( n - m)!) p  ^ {m} q  ^ {n-m} $
of the general term of the [[Binomial distribution|binomial distribution]], where

$$ 
\phi ( t)  = \ 
( pe  ^ {it} + q)  ^ {n}  = \ 
\sum _ {m = 0 } ^ { n } 
B ( m) e  ^ {itm}
$$

is the characteristic function of the binomial distribution. The expansion of  $   \mathop{\rm ln}  \phi ( t) $
in powers of  $  t $
yields a Gram–Charlier series of type  $  A $
for  $  B _ {0} ( x) $, 
whereas the expansion of  $   \mathop{\rm ln}  \phi ( t) $
in powers of  $  p $
yields a Gram–Charlier series of type  $  B $.

====References====
{|
|valign="top"|{{Ref|G}}|| J.P. Gram,   "Ueber die Entwicklung reeller Funktionen in Reihen mittelst der Methode der kleinsten Quadraten"  ''J. Reine Angew. Math.'' , '''94'''  (1883)  pp. 41–73
|-
|valign="top"|{{Ref|Ch}}|| C.V.L. Charlier,   "Frequency curves of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473037.png" /> in heterograde statistics"  ''Ark. Mat. Astr. Fysik'' , '''9''' :  25  (1914)  pp. 1–17
|-
|valign="top"|{{Ref|M}}|| A.K. Mitropol'skii,   "Curves of distributions" , Leningrad  (1960)  (In Russian)
|}

====Comments====
Cf. also [[Edgeworth series|Edgeworth series]].

====References====
{|
|valign="top"|{{Ref|Cr}}|| H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)  pp. Sect. 17.6
|}