Euler transform

In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.

Definition

The binomial transform, , of a sequence, , is the sequence defined by

$s_n = \sum_^n (-1)^k \binom a_k.$

Formally, one may write

$s_n = (Ta)_n = \sum_^n T_ a_k$

for the transformation, where is an infinite-dimensional operator with matrix elements .
The transform is an involution, that is,

$TT = 1$

or, using index notation,

$\sum_^\infty T_ T_ = \delta_$

where $\delta_$ is the Kronecker delta. The original series can be regained by

$a_n=\sum_^n (-1)^k \binom s_k.$

The binomial transform of a sequence is just the -th forward differences of the sequence, with odd differences carrying a negative sign, namely:

$\begin s_0 &= a_0 \\ s_1 &= - (\Delta a)_0 = -a_1+a_0 \\ s_2 &= (\Delta^2 a)_0 = -(-a_2+a_1)+(-a_1+a_0) = a_2-2a_1+a_0 \\ &\;\; \vdots \\ s_n &= (-1)^n (\Delta^n a)_0 \end$

where is the forward difference operator.

Some authors define the binomial transform with an extra sign, so that it is not self-inverse:

$t_n = \sum_^n (-1)^ \binom a_k$

whose inverse is

$a_n=\sum_^n \binom t_k.$

In this case the former transform is called the ''inverse binomial transform'', and the latter is just ''binomial transform''. This is standard usage for example in On-Line Encyclopedia of Integer Sequences.

Example

Both versions of the binomial transform appear in difference tables. Consider the following difference table:

Each line is the difference of the previous line. (The ''n''-th number in the ''m''-th line is ''a''_''m'',''n'' = 3^''n''−2(2^''m''+1''n''² + 2^''m''(1+6''m'')''n'' + 2^''m''-19''m''²), and the difference equation ''a''_{''m''+1,''n''} = ''a''_{''m'',''n''+1} - ''a''_''m'',''n'' holds.)

The top line read from left to right is = 0, 1, 10, 63, 324, 1485, ... The diagonal with the same starting point 0 is = 0, 1, 8, 36, 128, 400, ... is the noninvolutive binomial transform of .

The top line read from right to left is = 1485, 324, 63, 10, 1, 0, ... The cross-diagonal with the same starting point 1485 is = 1485, 1161, 900, 692, 528, 400, ... is the involutive binomial transform of .

Ordinary generating function

The transform connects the generating functions associated with the series. For the ordinary generating function, let

$f(x) = \sum_^\infty a_n x^n$

and

$g(x) = \sum_^\infty s_n x^n$

then

$g(x) = (Tf)(x) = \frac f.$

Euler transform

The relationship between the ordinary generating functions is sometimes called the Euler transform. It commonly makes its appearance in one of two different ways. In one form, it is used to accelerate the convergence of an alternating series. That is, one has the identity

$\sum_^\infty ^n a_n = \sum_^\infty ^n \frac$

which is obtained by substituting into the last formula above. The terms on the right hand side typically become much smaller, much more rapidly, thus allowing rapid numerical summation.

The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):

$\sum_^\infty ^n \binom a_n = \sum_^\infty ^n \binom \frac ,$

where .

The Euler transform is also frequently applied to the Euler hypergeometric integral $\,_2F_1$. Here, the Euler transform takes the form:

$\,_2F_1 (a,b;c;z) = (1-z)^ \,_2F_1 \left(c-a, b; c;\frac \right).$

ee for generalizations to other hypergeometric series.
The binomial transform, and its variation as the Euler transform, is notable for its connection to the continued fraction representation of a number. Let $0 < x < 1$ have the continued fraction representation

x= ;a_1, a_2, a_3,\cdots/math>

then

\frac = ;a_1-1, a_2, a_3,\cdots/math>

and

\frac = ;a_1+1, a_2, a_3,\cdots

Exponential generating function

For the exponential generating function, let

$\overline(x)= \sum_^\infty a_n \frac$

and

$\overline(x)= \sum_^\infty s_n \frac$

then

$\overline(x) = (T\overline)(x) = e^x \overline(-x).$

The Borel transform will convert the ordinary generating function to the exponential generating function.

Binomial convolution

Let $\$ and $\$, $n=0,1,2,\ldots,$ be sequences of complex numbers. Their binomial convolution is defined by
$(a\circ b)_n = \sum_^n \binom a_k b_,\ \ n=0,1,2,\ldots$
This convolution can be found in the book by R.L. Graham, D.E. Knuth and O. Patashnik: Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley (1989). It is easy to see that the binomial convolution is associative and commutative, and the sequence $\$ defined by $e_0 = 1$ and $e_n=0$ for $n=1,2,\ldots,$ serves as the identity under the binomial convolution. Further, it is easy to see that the sequences $\$ with $a_0\ne 0$ possess an inverse. Thus the set of sequences $\$ with $a_0\ne 0$ forms an Abelian group under the binomial convolution.

The binomial convolution arises naturally from the product of the exponential generating functions. In fact,
$\left( \sum^\infty_ a_n \frac \right) \left( \sum^\infty_ b_n \frac \right) = \sum^\infty_ (a\circ b)_n \frac.$

The binomial transform can be written in terms of binomial convolution. Let $\lambda_n = (-1)^n$ and $1_n=1$ for all $n$. Then
$(Ta)_n = (\lambda a\circ 1)_n.$
The formula
$t_n = \sum_^n ^ \binom a_k \iff a_n = \sum_^n \binom t_k$
can be interpreted as a Möbius inversion type formula
$t_n = (a\circ \lambda)_n \iff a_n = (t\circ 1)_n$
since $\lambda_n$ is the inverse of $1_n$ under the binomial convolution.

There is also another binomial convolution in the mathematical literature. The binomial convolution of arithmetical functions $f$ and $g$ is defined as
$(f\circ_B g)(n) = \sum_ \left( \prod_p \binom \right) f(d)g(n/d),$
where $n = \prod_p p^$ is the canonical factorization of a positive integer $n$ and $\binom$ is the binomial coefficient. This convolution appears in the book by P. J. McCarthy (1986) and was further studied by L. Toth and P. Haukkanen (2009).

Integral representation

When the sequence can be interpolated by a complex analytic function, then the binomial transform of the sequence can be represented by means of a Nörlund–Rice integral on the interpolating function.

Generalizations

Prodinger gives a related, modular-like transformation: letting

$u_n = \sum_^n \binom a^k ^ b_k$

gives

$U(x) = \frac B$

where and are the ordinary generating functions associated with the series $\$ and $\$, respectively.

The rising -binomial transform is sometimes defined as

$\sum_^n \binom j^k a_j.$

The falling -binomial transform is

$\sum_^n \binom j^ a_j.$

Both are homomorphisms of the kernel of the Hankel transform of a series.

In the case where the binomial transform is defined as

$\sum_^n ^ \binom a_i = b_n.$

Let this be equal to the function $\mathfrak J(a)_n=b_n.$

If a new forward difference table is made and the first elements from each row of this table are taken to form a new sequence $\$, then the second binomial transform of the original sequence is,

$\mathfrak J^2(a)_n=\sum_^n(-2)^\binoma_i.$

If the same process is repeated ''k'' times, then it follows that,

$\mathfrak J^k(a)_n=b_n=\sum_^n(-k)^\binoma_i.$

Its inverse is,

$\mathfrak J^(b)_n = a_n=\sum_^n k^\binomb_i.$

This can be generalized as,

$\mathfrak J^k(a)_n = b_n = (\mathbf E-k)^na_0$

where $\mathbf E$ is the shift operator.

Its inverse is

$\mathfrak J^(b)_n = a_n = (\mathbf E+k)^nb_0.$

See also

* Newton series
* Hankel matrix
* Möbius transform
* Stirling transform
* Euler summation
* Binomial QMF
* Riemann–Liouville integral
* List of factorial and binomial topics

References

* John H. Conway and Richard K. Guy, 1996, ''The Book of Numbers''
* Donald E. Knuth, ''The Art of Computer Programming Vol. 3'', (1973) Addison-Wesley, Reading, MA.
* Helmut Prodinger
Some information about the binomial transform
The Fibonacci Quarterly 32 (1994), 412-415.
*
* {{cite journal, last1=Borisov, first1=B., last2=Shkodrov , first2=V., year=2007, title=Divergent Series in the Generalized Binomial Transform, journal=Adv. Stud. Cont. Math., volume=14, number=1, pages=77–82, url=https://www.kci.go.kr/kciportal/ci/sereArticleSearch/ciSereArtiView.kci?sereArticleSearchBean.artiId=ART001258138
* Khristo N. Boyadzhiev, ''Notes on the Binomial Transform'', Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.
* R.L. Graham, D.E. Knuth and O. Patashnik: Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley (1989).
* P. J. McCarthy, Introduction to Arithmetical Functions, Springer-Verlag, 1986.
* P. Haukkanen, On a binomial convolution of arithmetical functions, Nieuw Arch. Wisk. (IV) 14 (1996), no. 2, 209--216.
* L. Toth and P. Haukkanen, On the binomial convolution of arithmetical functions, J. Combinatorics and Number Theory 1(2009), 31-48.
* P. Haukkanen, Some binomial inversions in terms of ordinary generating functions. Publ. Math. Debr. 47, No. 1-2, 181-191 (1995).

External links

on Wolfram MathWorld
Binomial transform
in the OEIS wiki

Transforms
Factorial and binomial topics
Hypergeometric functions