DD-finite functions in Sage 2 Using the package Computing beyond - PowerPoint PPT Presentation

Installation and usage Computing and proving Conclusions Installation and usage Computing and proving Conclusions Outline 1 Installing the package DD-finite functions in Sage 2 Using the package Computing beyond holonomic functions 3 Computing with the package Antonio Jiménez-Pastor 4 Proving with the package 5 Conclusions FPSAC (Jul. 2019) DD-finite functions in Sage DD-finite functions in Sage Installation and usage Computing and proving Conclusions Installation and usage Computing and proving Conclusions How to install the package DD-finite Functions Git repository Definition http://git.risc.jku.at/gitweb/?p=ajpastor/diff_defined_functions.git Let f ∈ K [[ x ]] and R ⊂ K [[ x ]] a ring. We say that f is differentially definable over R if there exist d ∈ N and elements in Zip from webpage R r 0 ( x ) , ..., r d ( x ) such that: https://www.dk-compmath.jku.at/Members/antonio/sage-package-dd_functions r d ( x ) f ( d ) ( x ) + ... + r 0 ( x ) f ( x ) = 0 . PyPi (in process) pip install dd_functions Stable version Easy to update DD-finite functions in Sage DD-finite functions in Sage

Installation and usage Computing and proving Conclusions Installation and usage Computing and proving Conclusions D-finite examples DD-finite examples Elementary functions Classic non-holonomic Double exponential ( e e x − 1 ), tangent ( Tan ),... Exponential ( Exp ), trigonometric ( Sin , Cos ),... Special functions Mathieu functions Bessel functions ( BesselD ), hypergeometric functions DD-finite generalization of the sine and cosine ( HypergeometricFunction ),... Combinatorial functions Generating functions for holonomic sequences (Catalan numbers, Fibonacci sequence, etc) DD-finite functions in Sage DD-finite functions in Sage Installation and usage Computing and proving Conclusions Installation and usage Computing and proving Conclusions Operations supported Extracting sequence Ordinary generating functions Method getSequenceElement allows to get the associated Arithmetic operations sequence. Addition ( + , - ), product ( * , / , ˆ ) Exponential generating functions Differential operations Method getInitialValue allows to get the associated Derivative ( derivative ), integration ( integrate ) exponential sequence. Composition Using the standard call in Sage. DD-finite functions in Sage DD-finite functions in Sage

Installation and usage Computing and proving Conclusions Installation and usage Computing and proving Conclusions Proving DD-finite identities Proving DD-finite identities Constant Wronskian Constant Wronskian w ′′ ( x ) − ( a − 2 q cos(2 x )) w ( x ) = 0, w ′′ ( x ) − ( a − 2 q cos(2 x )) w ( x ) = 0, w ′ 1 w 2 − w ′ 2 w 1 = 1. w ′ 1 w 2 − w ′ 2 w 1 = 1. DD-finite functions in Sage DD-finite functions in Sage Installation and usage Computing and proving Conclusions Installation and usage Computing and proving Conclusions Conclusions Thank you! Features Sage package for DD-finite functions Arithmetic and differentil porperties implemented Antonio Jiménez-Pastor Composition implemented https://www.dk-compmath.jku.at/people/antonio Zero recognition (equality) implemented https://www.risc.jku.at/home/ajpastor To be done Sage package: Improve performance https://www.dk-compmath.jku.at/Members/antonio/ Getting more examples sage-package-dd_functions Polish the current implementation DD-finite functions in Sage DD-finite functions in Sage