Eisenstein Series - Recurrence Relation

Recurrence Relation

Any holomorphic modular form for the modular group can be written as a polynomial in and . Specifically, the higher order 's can be written in terms of and through a recurrence relation. Let . Then the satisfy the relation

for all . Here, is the binomial coefficient and and .

The occur in the series expansion for the Weierstrass's elliptic functions:

\wp(z) =\frac{1}{z^2} + z^2 \sum_{k=0}^\infty \frac {d_k z^{2k}}{k!} =\frac{1}{z^2} + \sum_{k=1}^\infty (2k+1) G_{2k+2} z^{2k}

Read more about this topic:  Eisenstein Series

Famous quotes containing the words recurrence and/or relation:

    “Forgetfulness is necessary to remembrance. Ideas are retained by renovation of that impression which time is always wearing away, and which new images are striving to obliterate. If useless thoughts could be expelled from the mind, all the valuable parts of our knowledge would more frequently recur, and every recurrence would reinstate them in their former place.”
    —Samuel Johnson (1709–1784)

    “The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
    —Ralph Waldo Emerson (1803–1882)