Lambert series and Liouville's identities

Rank-crank Type Pdes and Generalized Lambert Series Identities

We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watso...

Lambert series and conjugacy classes in GL

Algebraic independence of some Lambert series

1] proved that if Y k0 (1 0 z F k) = X k0 "(k)z k ; then "(k) = 0 or 61 for any k 0. Tamura [7] generalized this result by proving the following theorem: Let fR k g k0 be a linear recurrence of positive integers satisfying R k+n = R k+n01 + 1 1 1 + R k (k 0) with n 2 and let P(z) = Y k0 (1 0 z R k) = X k0 "(k)z k : Then, if n is even, f"(k) j k 0g is a nite set; if in addition R k = 2 k (0 k n0...

Little q-Legendre polynomials and irrationality of certain Lambert series

Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős [9]. In 1991–1992 Peter Borwein [4] [5] used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to f...

q–ENGEL SERIES EXPANSIONS AND SLATER’S IDENTITIES

We describe the q–Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater’s list. In our study Al-Salam/Ismail and Santos polynomials play a crucial rôle. Dedicated to the memory of John Knopfmacher 1937–1999