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Euler Transform


There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.

The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series

 sum_(k=0)^infty(-1)^ka_k=a_0-a_1+a_2-...
(1)

into a series with more rapid convergence to the same value to

 s=sum_(k=0)^infty((-1)^kDelta^ka_0)/(2^(k+1)),
(2)

where the forward difference is defined by

 Delta^ka_0=sum_(m=0)^k=(-1)^m(k; m)a_(k-m)
(3)

(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the fact that when z=-1 is taken in the second of Euler's hypergeometric transformations

 _2F_1(a,b;c;z)=(_2F_1(c-a,b;c;z/(z-1)))/((1-z)^b),
(4)

where _2F_1(a,b,;c;z) is a hypergeometric function, it gives Euler's convergence improvement transformation of the series _2F_1(a,b;c;-1) (Abramowitz and Stegun 1972, p. 555).

The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21). If a_1, a_2, ... and b_1, b_2, ... are related by

 1+sum_(n=1)^inftyb_nx^n=product_(i=1)^infty1/((1-x^i)^(a_i))
(5)

or, in terms of generating functions A(x) and B(x),

 1+B(x)=exp[sum_(k=1)^infty(A(x^k))/k],
(6)

then {b_n} is said to be the Euler transform of {a_n} (Sloane and Plouffe 1995, p. 20). The Euler transform can be effected by introducing the intermediate series c_1, c_2, ... given by

 c_n=sum_(d|n)da_d,
(7)

then

 b_n=1/n[c_n+sum_(k=1)^(n-1)c_kb_(n-k)],
(8)

with b_1=c_1. Similarly, the inverse transform can be effected by computing the intermediate series as

 c_n=nb_n-sum_(k=1)^(n-1)c_kb_(n-k),
(9)

then

 a_n=1/nsum_(d|n)mu(n/d)c_d,
(10)

where mu(n) is the Möbius function.

In graph theory, if a_n is the number of unlabeled connected graphs on n nodes satisfying some property, then b_n is the total number of unlabeled graphs (connected or not) with the same property. This application of the Euler transform is called Riddell's formula for unlabeled graph (Sloane and Plouffe 1995, p. 20).

There are also important number theoretic applications of the Euler transform. For example, if there are a_1 kinds of parts of size 1, a_2 kinds of parts of size 2, etc., in a given type of partition, then the Euler transform b_n of a_n is the number of partitions of n into these integer parts. For example, if a_n=1 for all n, then b_n is the number of partitions of n into integer parts. Similarly, if a_n=1 for n prime and a_n=0 for n composite, then b_n is the number of partitions of n into prime parts (Sloane and Plouffe 1995, p. 21). Other applications are given by Andrews (1986), Andrews and Baxter (1989), and Cameron (1989).


See also

Binomial Transform, Convergence Improvement, Euler's Hypergeometric Transformations, Forward Difference, Integer Sequence, Möbius Transform, Riddell's Formula, Stirling Transform, Wynn's Epsilon Method

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986.Andrews, G. E. and Baxter, R. J. "A Motivated Proof of the Rogers-Ramanujan Identities." Amer. Math. Monthly 96, 401-409, 1989.Beeler, M. et al. Item 120 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 55, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/series.html#item120.Bernstein, M. and Sloane, N. J. A. "Some Canonical Sequences of Integers." Linear Algebra Appl. 226//228, 57-72, 1995.Cameron, P. J. "Some Sequences of Integers." Disc. Math. 75, 89-102, 1989.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1163, 1980.Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 20-21, 1995.

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Euler Transform

Cite this as:

Weisstein, Eric W. "Euler Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerTransform.html

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