Binomial transform

The binomial transform, T, of a sequence, {a_n}, is the sequence {s_n} defined by

${\displaystyle s_{n}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}a_{k}.}$

Formally, one may write

${\displaystyle s_{n}=(Ta)_{n}=\sum _{k=0}^{n}T_{nk}a_{k}}$

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

${\displaystyle TT=1}$

or, using index notation,

${\displaystyle \sum _{k=0}^{\infty }T_{nk}T_{km}=\delta _{nm}}$

where ${\displaystyle \delta _{nm}}$ is the Kronecker delta. The original series can be regained by

${\displaystyle a_{n}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}s_{k}.}$

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

${\displaystyle {\begin{aligned}s_{0}&=a_{0}\\s_{1}&=-(\Delta a)_{0}=-a_{1}+a_{0}\\s_{2}&=(\Delta ^{2}a)_{0}=-(-a_{2}+a_{1})+(-a_{1}+a_{0})=a_{2}-2a_{1}+a_{0}\\&\;\;\vdots \\s_{n}&=(-1)^{n}(\Delta ^{n}a)_{0}\end{aligned}}}$

where Δ is the forward difference operator.

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

${\displaystyle t_{n}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}a_{k}}$

whose inverse is

${\displaystyle a_{n}=\sum _{k=0}^{n}{n \choose k}t_{k}.}$

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.

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 = 3ⁿ⁻²(2^m+1n² + 2^m(1+6m)n + 2^m-19m²), 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 {a_n} = 0, 1, 10, 63, 324, 1485, ... The diagonal with the same starting point 0 is {t_n} = 0, 1, 8, 36, 128, 400, ... {t_n} is the noninvolutive binomial transform of {a_n}.

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

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

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}}$

and

${\displaystyle g(x)=\sum _{n=0}^{\infty }s_{n}x^{n}}$

then

${\displaystyle g(x)=(Tf)(x)={\frac {1}{1-x}}f\left({\frac {-x}{1-x}}\right).}$

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

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(\Delta ^{n}a)_{0}}{2^{n+1}}}}$

which is obtained by substituting x = 1/2 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):

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{n+p \choose n}a_{n}=\sum _{n=0}^{\infty }(-1)^{n}{n+p \choose n}{\frac {(\Delta ^{n}a)_{0}}{2^{n+p+1}}},}$

where p = 0, 1, 2,…

The Euler transform is also frequently applied to the Euler hypergeometric integral ${\displaystyle \,_{2}F_{1}}$. Here, the Euler transform takes the form:

${\displaystyle \,_{2}F_{1}(a,b;c;z)=(1-z)^{-b}\,_{2}F_{1}\left(c-a,b;c;{\frac {z}{z-1}}\right).}$

[See [1] 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 ${\displaystyle 0<x<1}$ have the continued fraction representation

${\displaystyle x=[0;a_{1},a_{2},a_{3},\cdots ]}$

then

${\displaystyle {\frac {x}{1-x}}=[0;a_{1}-1,a_{2},a_{3},\cdots ]}$

and

${\displaystyle {\frac {x}{1+x}}=[0;a_{1}+1,a_{2},a_{3},\cdots ].}$

For the exponential generating function, let

${\displaystyle {\overline {f}}(x)=\sum _{n=0}^{\infty }a_{n}{\frac {x^{n}}{n!}}}$

and

${\displaystyle {\overline {g}}(x)=\sum _{n=0}^{\infty }s_{n}{\frac {x^{n}}{n!}}}$

then

${\displaystyle {\overline {g}}(x)=(T{\overline {f}})(x)=e^{x}{\overline {f}}(-x).}$

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

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.

Prodinger gives a related, modular-like transformation: letting

${\displaystyle u_{n}=\sum _{k=0}^{n}{n \choose k}a^{k}(-c)^{n-k}b_{k}}$

gives

${\displaystyle U(x)={\frac {1}{cx+1}}B\left({\frac {ax}{cx+1}}\right)}$

where U and B are the ordinary generating functions associated with the series ${\displaystyle \{u_{n}\}}$ and ${\displaystyle \{b_{n}\}}$, respectively.

The rising k-binomial transform is sometimes defined as

${\displaystyle \sum _{j=0}^{n}{n \choose j}j^{k}a_{j}.}$

The falling k-binomial transform is

${\displaystyle \sum _{j=0}^{n}{n \choose j}j^{n-k}a_{j}}$.

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

In the case where the binomial transform is defined as

${\displaystyle \sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}a_{i}=b_{n}.}$

Let this be equal to the function ${\displaystyle {\mathfrak {J}}(a)_{n}=b_{n}.}$

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 ${\displaystyle \{b_{n}\}}$, then the second binomial transform of the original sequence is,

${\displaystyle {\mathfrak {J}}^{2}(a)_{n}=\sum _{i=0}^{n}(-2)^{n-i}{\binom {n}{i}}a_{i}.}$

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

${\displaystyle {\mathfrak {J}}^{k}(a)_{n}=b_{n}=\sum _{i=0}^{n}(-k)^{n-i}{\binom {n}{i}}a_{i}.}$

Its inverse is,

${\displaystyle {\mathfrak {J}}^{-k}(b)_{n}=a_{n}=\sum _{i=0}^{n}k^{n-i}{\binom {n}{i}}b_{i}.}$

This can be generalized as,

${\displaystyle {\mathfrak {J}}^{k}(a)_{n}=b_{n}=(\mathbf {E} -k)^{n}a_{0}}$

where ${\displaystyle \mathbf {E} }$ is the shift operator.

Its inverse is

${\displaystyle {\mathfrak {J}}^{-k}(b)_{n}=a_{n}=(\mathbf {E} +k)^{n}b_{0}.}$

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

• 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, 1992, Some information about the Binomial transform Archived 2007-03-12 at the Wayback Machine
• Spivey, Michael Z.; Steil, Laura L. (2006). "The k-Binomial Transforms and the Hankel Transform". Journal of Integer Sequences. 9: 06.1.1. Bibcode:2006JIntS...9...11S.
• Borisov, B.; Shkodrov, V. (2007). "Divergent Series in the Generalized Binomial Transform". Adv. Stud. Cont. Math. 14 (1): 77–82.
• Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.

• Binomial Transform
• Transformations of Integer Sequences