The scientific output achieved in 2005 by the SFB project group F1305 is documented in the form of 21 publications: 8 articles were published in journals and 4 in conference proceedings; 9 technical reports have been produced, 4 of which are already accepted for journal publication. Additionally, two PhD-theses [9,14] have been completed.

# 1. Identities

Various refined summation algorithms [20,21,22,23,24,26] have been developed by Schneider that enable one to simplify and/or evaluate complicated multi-sum expressions. Examples of successful applications of these tools are: A computer proof of the "Totally symmetric plane partition" theorem [1], a quadruple sum expression that evaluates to zeta-functions [19,25], proofs of identities that are needed for Padé approximation [5,6], and the derivation of reciprocity laws of harmonic numbers that arise in the analysis of algorithms [18]. In addition, recurrences were computed that could speed up the computations in Finite Element Methods [2]; see project F1301.

Kauers and Schneider extended the summation algorithms of Schneider by allowing generic (unspecified) sequences within sums [17]. In [16] they illustrate how these algorithms can be used to discover new general identities.

In a joint effort, Paule, Gerhold, Kauers, Schneider, and Zimmermann could provide proofs of various identities in the Handbook of Mathematical Functions (Abramowitz/Stegun) whose original proofs have been lost [12]. One of of these identities is shown in the box below.

In his Ph.D. thesis [14], Kauers presents a collection of algorithms for sequences which can be defined by nonlinear higher order difference equations. An implementation of these algorithms in form of a Mathematica package [15] is able to prove and to discover identities which were previously considered out of scope of symbolic computation. Examples include properties of Somos sequences, nested C-finite expressions, orthogonal polynomials, continued fractions, etc. A large collection of example applications is included in the thesis.

# 2. Inequalities and Asymptotics

Gerhold and Kauers have proposed a procedure for automatically proving inequalities among expressions that are defined via recurrence equations [10]. With this procedure, it was possible to verify a large number of inequalities appearing in the literature by a computer procedure for the first time. A remarkable example is the computer proof of Turán's inequality for Legendre polynomials [11].

Inequalities are in general harder than identities, and the procedure of Gerhold and Kauers is not able to provide a proof for every true inequality to which it is applicable. A conjectured inequality which arose in the numerical work of J. Schöberl (F1319), for example, is in the right shape for the method of Gerhold/Kauers to apply, but the method fails to supply a proof. Attempts to prove this inequality by hand using asymptotic arguments have also failed so far. See [13] for some work that has been done by Gerhold, Kauers, and Schöberl on this inequality.

In joint work with J.P. Bell [3], Gerhold has obtained a fairly satisfactory result about the sign of oscillating linear recurrence sequences: If a C-finite sequence has no real positive dominating root, then its positivity set and its negativity set both have positive density. Moreover, the density can assume each value from ]0, 1[.

# 3. Non-Holonomicity

Non-holonomicity results give some evidence on the algorithmic complexity of a sequence, since values of holonomic sequences can be readily computed by the linear recurrence with polynomial coefficients that defines them. Flajolet, Gerhold, and Salvy [8] have shown by an asymptotic method [7] that the sequence is not holonomic. Along the way, amusing asymptotic results like

# Bibliography

- 1
- ANDREWS, G., PAULE, P.,
AND SCHNEIDER, C.

Plane Partitions VI: Stembridges TSPP Theorem.

*Adv. in Appl. Math. 34*, 4 (2005), 709-739. - 2
- BECIROVIC, A., PAULE, P.,
PILLWEIN, V., RIESE, A.,
SCHNEIDER, C., AND
SCHÖBERL, J.

Hypergeometric summation algorithms for high order finite elements.

SFB-Report 2006-8, J. Kepler Universität, Linz, 2006.

Submitted. - 3
- BELL, J. P., AND
GERHOLD, S.

The Positivity Set of a Recurrence Sequence.

*Israel J. Math.*(2005).

To appear. - 4
- BELL, J. P.,
GERHOLD, S., KLAZAR, M.,
AND LUCA, F.

Non-holonomicity proofs for sequences defined via elementary functions.

Submitted. - 5
- DRIVER, K., PRODINGER,
H., SCHNEIDER, C., AND
WEIDEMAN, A.

Padé approximations to the logarithm III: Alternative methods and additional results.

*To appear in Ramanujan J.*(2006). - 6
- DRIVER, K., PRODINGER,
H., SCHNEIDER, C., AND
WEIDEMAN, J. A. C.

Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation.

*Ramanujan J. 11*, 2 (2006). - 7
- FLAJOLET, P., GERHOLD,
S., AND SALVY, B.

On the non-holonomic character of logarithms, powers and the th prime function.

*Electronic Journal of Combinatorics 11*, 2 (2005), 1-16. - 8
- FLAJOLET, P., GERHOLD,
S., AND SALVY, B.

Asymptotic analysis of the generating functions of certain closed form sequences.

In preparation, 2006. - 9
- GERHOLD, S.

*Combinatorial Sequences: Non-Holonomicity and Inequalities*.

PhD thesis, RISC, J. Kepler Universität, Linz, 2005. - 10
- GERHOLD, S., AND
KAUERS, M.

A procedure for proving special function inequalities involving a discrete parameter.

In*Proceedings of ISSAC'05*(2005), pp. 156-162. - 11
- GERHOLD, S., AND
KAUERS, M.

A computer proof of Turán's inequality.

*Journal of Inequalities in Pure and Applied Mathematics 7*, 2 (2006). - 12
- GERHOLD, S., KAUERS, M.,
OLVER, F., PAULE, P.,
SCHNEIDER, C., AND
ZIMMERMANN, B.

Computer proofs of some identities for bessel functions of fractional order.

In preparation, 2006. - 13
- GERHOLD, S., KAUERS, M.,
AND SCHÖBERL, J.

On a conjectured inequality for a sum of Legendre polynomials.

SFB-Report 2006-11, J. Kepler Universität, Linz, 2006. - 14
- KAUERS, M.

*Algorithms for Nonlinear Higher Order Difference Equations*.

PhD thesis, RISC, J. Kepler Universität, Linz, 2005. - 15
- KAUERS, M.

SumCracker -- a package for manipulating symbolic sums and related objects.

SFB-Report 2005-21, J. Kepler Universität, Linz, 2005.

Submitted. - 16
- KAUERS, M., AND
SCHNEIDER, C.

Application of unspecified sequences in symbolic summation.

*To appear in ISSAC'06*(2006). - 17
- KAUERS, M., AND
SCHNEIDER, C.

Indefinite summation with unspecified sequences.

*To appear in Discrete Math.*(2006). - 18
- KUBA, M., PRODINGER, H.,
AND SCHNEIDER, C.

Generalized reciprocity laws for sums of harmonic numbers.

SFB-Report 2005-17, J. Kepler Universität, Linz, 2005.

Submitted. - 19
- PEMANTLE, R., AND
SCHNEIDER, C.

When is 0.999... equal to 1?

*To appear in Amer. Math. Monthly*(2006). - 20
- SCHNEIDER, C.

Degree bounds to find polynomial solutions of parameterized linear difference equations in -fields.

*Appl. Algebra Engrg. Comm. Comput. 16*, 1 (2005), 1-32. - 21
- SCHNEIDER, C.

Finding telescopers with minimal depth for indefinite nested sum and product expressions.

In*Proceedings of ISSAC'05*(2005), M. Kauers, Ed., ACM, pp. 285-292. - 22
- SCHNEIDER, C.

A new Sigma approach to multi-summation.

*Adv. in Appl. Math. 34*, 4 (2005), 740-767. - 23
- SCHNEIDER, C.

Product representations in -fields.

*Ann. Comb. 9*, 1 (2005), 75-99. - 24
- SCHNEIDER, C.

Solving parameterized linear difference equations in terms of indefinite nested sums and products.

*J. Differ. Equations Appl. 11*, 9 (2005), 799-821. - 25
- SCHNEIDER, C.

Some Notes On "When is 0.999... equal to 1?".

In*Mathematics, Algorithms, Proofs*(2005), no. 05021 in Dagstuhl Seminar Proceedings. - 26
- SCHNEIDER, C.

Simplifying sums in -extensions.

SFB-Report 2006-13, J. Kepler Universität, Linz, 2006.

Submitted.

SpezialForschungsBereich SFB F013 | Special Research Program of the FWF - Austrian Science Fund