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

edit

The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined by

 

Formally, one may write

 

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

 

or, using index notation,

 

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

 

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

 

where Δ is the forward difference operator.

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

 

whose inverse is

 

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

edit

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

0   1   10   63   324   1485
  1   9   53   261   1161
    8   44   208   900
      36   164   692
        128   528
          400

Each line is the difference of the previous line. (The n-th number in the m-th line is am,n = 3n−2(2m+1n2 + 2m(1+6m)n + 2m-19m2), and the difference equation am+1,n = am,n+1 - am,n holds.)

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

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

Ordinary generating function

edit

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

 

and

 

then

 

Euler transform

edit

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

 

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):

 

where p = 0, 1, 2,…

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

 

[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   have the continued fraction representation

 

then

 

and

 

Exponential generating function

edit

For the exponential generating function, let

 

and

 

then

 

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


Binomial convolution

edit

Let   and  ,   be sequences of complex numbers. Their binomial convolution is defined by

 

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   and   for   serves as the identity under the binomial convolution. Further, it is easy to see that the sequences   with   possess an inverse. Thus the set of sequences   with   forms an Abelian group under the binomial convolution.

The binomial convolution arises naturally from the product of the exponential generating functions. In fact,

 


The binomial transform can be written in terms of binomial convolution. Let   and   for all  . Then

 

The formula

 

can be inerpreted as a Möbius inversion type formula

 

since   is the inverse of   under the binomial convolution.


There is also another binomial convolution in the mathematical literature. The binomial convolution of arithmetical functions   and   is defined as

 

where   is the canonical factorization of a positive integer   and   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

edit

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

edit

Prodinger gives a related, modular-like transformation: letting

 

gives

 

where U and B are the ordinary generating functions associated with the series   and  , respectively.

The rising k-binomial transform is sometimes defined as

 

The falling k-binomial transform is

 .

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

In the case where the binomial transform is defined as

 

Let this be equal to the function  

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,

 

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

 

Its inverse is,

 

This can be generalized as,

 

where   is the shift operator.

Its inverse is

 

See also

edit

References

edit
  1. ^ Miller, Allen R.; Paris, R. B. (2010). "Euler-type transformations for the generalized hypergeometric function". Z. Angew. Math. Phys. 62 (1): 31–45. doi:10.1007/s00033-010-0085-0. S2CID 30484300.
  • 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.
  • 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).
edit