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: 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 For the exponential generating function, let and then The Borel transform will convert the ordinary generating function to the exponential generating function. 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 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 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, 1992, Some information about the Binomial transform Michael Z. Spivey and Laura L. Steil, 2006, The k-Binomial Transforms and the Hankel Transform Borisov B. and 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. External links Binomial Transform Transformations of Integer Sequences Transforms