Symbolic summation in difference fields has been introduced by Karr’s groundbreaking work karr1981summation ; karr1985theory . He defined the so-called -fields which are composed by a field and a field automorphism . Here the field is built by a tower of transcendental field extensions whose generators either represent sums or products where the summands or multiplicands are elements from the field below. In particular, the following problem has been solved: given such a -field and given . Decide algorithmically, if there exists a with
Hence if and can be rephrased to expressions and in terms of indefinite nested sums and products, one obtains the telescoping relation
Then summing this telescoping equation over a valid range, say , one gets the
In a nutshell, the following strategy can be applied: (I) construct an appropriate -field in which a given summand in terms of indefinite nested sums and products is rephrased by ; (II) compute such that (1) holds; (III) rephrase to an expression such that (2) holds.
In the last years various new algorithms and improvements of Karr’s difference field theory have been developed in order to obtain a fully automatic simplification machinery for nested sums. Here the key observation is that a sum can be either expressed in the existing difference field by solving the telescoping problem (2) or –if this is not possible– it can be adjoined as a new extension on top of the already constructed field yielding again a -field; see Theorem 2.1(3) below. By a careful construction of one can simplify sum expressions such that the nesting depth is minimized Schneider:08c , or the number Schneider:15 or the degree Schneider:07d of the objects arising in the summands are optimized.
In contrast to sums, representing products in -fields is not possible in general. In particular, the alternating sign , which arises frequently in applications, can be represented properly only in a ring with zero divisors introducing relations such as . In schneider2005product and a streamlined version worked out in Schneider:14 , this situation has been cured for the class of hypergeometric products of the form with and being a rational function with coefficient from the rational numbers: namely, a finite number of such products can be always represented in a -field adjoined with the element . In particular, nested sums defined over such products can be formulated automatically in difference rings built by the so-called -extensions schneider2016difference ; schneider2017summation . This means that the difference rings are constructed by transcendental ring extensions and algebraic ring extensions with generators of the form where is a primitive root of unity. Within this setting schneider2016difference ; schneider2017summation , one can then solve the telescoping problem for indefinite sums (see Eq. (1)) and more generally the creative telescoping problem petkovvsek1996b to compute linear recurrences for definite sums. Furthermore one can simplify the so-called d’Alembertian Petkov:92 ; Abramov:94 ; Abramov:96 ; vanHoeij:99 or Liouvillian solutions van2006galois ; Petkov:2013 of linear recurrences which are given in terms of nested sums defined over hypergeometric products. For many problems coming, e.g., from combinatorics or particle physics (for the newest applications see Sulzgruber:16 or Schneider:16b ) this difference ring machinery with more than 100 extension variables works fine. But in more general cases, one is faced with nested sums defined not only over hypergeometric but also over mixed multibasic products. Furthermore, these products might not be expressible in but only in an algebraic number field, i.e., in a finite algebraic field extension of .
Let be a rational function field over a field and let be a rational function field over . is a mixed -multibasic hypergeometric product in , if and is chosen big enough (see Ex. 2.9 below) such that has no pole and is non-zero for all with . If which is free of , then is called a -multibasic hypergeometric product in . If , then it is called a basic or -hypergeometric product in where . If and , then is called a hypergeometric product in . Finally, if , it is called constant or geometric product in .
Let denote and denote . Further, we define the set of ground expressions111Their elements are considered as expressions that can be evaluated for sufficiently large . , and . Moreover, we define with as the set of all such products where the multiplicand is taken from . Finally, we introduce the set of product expressions as the set of all elements
with , finite, and .
For this class where the subfield of itself can be a rational function field over an algebraic number field, we will solve the following problem.
innertopmargin = 10pt,
Problem RPE: Representation of Product Expressions.
Let . Given ;
find with a finite algebraic field extension222If is a rational function field over an algebraic number field , then in worst case is extended to where is an algebraic extension of . Subsequently, all algebraic field extensions are finite. of and a natural number with the following properties:
for all with ;
The product expressions in (apart from products over roots of unity) are algebraically independent among each other.
The zero-recognition property holds, i.e., holds for all from a certain point on if and only if is the zero-expression.
Internally, the multiplicands of the products are factorized and the monic irreducible factors, which are shift-equivalent, are rewritten in terms of one of these factors; compare abramov1971summation ; Paule:95 ; schneider2005product ; Petkov:10 ; ZimingLi:11 . Then using results of schneider2010parameterized ; Singer:08 we can conclude that products defined over these irreducible factors can be rephrased as transcendental difference ring extensions. Using similar strategies, one can treat the content coming from the monic irreducible polynomials, and obtains finally an -extension in which the products can be rephrased. We remark that the normal forms presented in ZimingLi:11 are closely related to this representation and enable one to check, e.g., if the given products are algebraically independent. Moreover, there is an algorithm KZ:08 that can compute all algebraic relations for -finite sequences, i.e., it finds certain ideals from whose elements evaluate to zero. Our main focus is different. We will compute alternative products which are by construction algebraically independent among each other and which enable one to express the given products in terms of the algebraic independent products. In particular, we will make this algebraic independence statement (see property (2) of 1.1) very precise by embedding the constructed -extension explicitly into the ring of sequences petkovvsek1996b by using results from schneider2017summation . The derived algorithms implemented in Ocansey’s Mathematica package NestedProducts supplement the summation package Sigma Sigma and enable one to formulate nested sums over such general products in the setting of -extensions. As a consequence, it is now possible to apply completely automatically the summation toolbox Petkov:10 ; bronstein2000solutions ; karr1981summation ; schneider2005product ; Schneider:08c ; Schneider:07d ; schneider2010parameterized ; Sigma ; Schneider:14 ; Schneider:15 ; schneider2016difference ; schneider2016symbolic ; schneider2017summation for simplification of indefinite and definite nested sums defined over such products.
The outline of the article is as follows. In Section 2 we define the basic notions of -extensions and present the main results to embed a difference ring built by -extensions into the ring of sequences. In Section 3 our 1.1 is reformulated to Theorem 3.1 in terms of these notions, and the basic strategy of how this problem will be tackled is presented. In Section 4 the necessary properties of the constant field are worked out that enable one to execute our proposed algorithms. Finally, in Sections 5 and 6 the hypergeometric case and afterwards the mixed multibasic case are treated. A conclusion is given in Section 7.
2 Ring of sequences, difference rings and difference fields
In this section, we discuss the algebraic setting of difference rings (resp. fields) and the ring of sequences as they have been elaborated in karr1981summation ; schneider2016difference ; schneider2017summation . In particular, we demonstrate how sequences generated by expressions in (resp. ) can be modeled in this algebraic framework.
2.1 Difference fields and difference rings
A difference ring (resp. field) is a ring (resp. field) together with a ring (resp. field) automorphism . Subsequently, all rings (resp. fields) are commutative with unity; in addition they contain the set of rational numbers , as a subring (resp. subfield). The multiplicative group of units of a ring (resp. field) is denoted by . A ring (resp. field) is computable if all of it’s operations are computable. A difference ring (resp. field) is computable if and are both computable. Thus, given a computable difference ring (resp. field), one can decide if . The set of all such elements for a given difference ring (resp. field) denoted by
forms a subring (resp. subfield) of . In this article, will always be a field called the constant field of . Note that it contains as a subfield. For any difference ring (resp. field) we shall denote the constant field by .
The construction of difference rings/fields will be accomplished by a tower of difference ring/field extensions. A difference ring is said to be a difference ring extension of a difference ring if is a subring of and for all , (i.e., ). The definition of a difference field extension is the same by only replacing the word ring with field. In the following we do not distinguish anymore between and .
In the following we will consider two types of product extensions. Let be a difference ring (in which products have already been defined by previous extensions). Let be a unit and consider the ring of Laurent polynomials (i.e., is transcendental over ). Then there is a unique difference ring extension of with and . The extension here is called a product-extension (in short -extension) and the generator is called a -monomial. Suppose that is a field and is a rational function field (i.e., is transcendental over ). Let . Then there is a unique difference field extension of with . We call the extension a -field extension and a -monomial. In addition, we get the chain of extensions .
Furthermore, we consider extensions which model algebraic objects like where is a -th root of unity for some with . Let be a difference ring and let be a primitive -th root of unity, (i.e., and is minimal). Take the difference ring extension of with being transcendental over and . Note that this construction is also unique. Consider the ideal and the quotient ring . Since is closed under and i.e., is a reflexive difference ideal, we have a ring automorphism defined by . In other words, is a difference ring. Note that by this construction the ring can naturally be embedded into the ring by identifying with , i.e., . Now set . Then is a difference ring extension of subject to the relations and . This extension is called an algebraic extension (in short -extension) of order . The generator, is called an -monomial and we define as its order. Note that the -monomial , with the relations and models with the relations and . In addition, the ring is not an integral domain (i.e., it has zero-divisors) since but .
We introduce the following notations for convenience. Let be a difference ring extension of with . denotes the ring of Laurent polynomials (i.e., is transcendental over ) if is a -extension of . Lastly, denotes the ring with but subject to the relation if is an -extension of of order . We say that the difference ring extension of is an -extension (and is an -monomial) if it is an - or a -extension. Finally, we call a (nested) -extension/-extension of it is built by a tower of such extensions.
Throughout this article, we will restrict ourselves to the following classes of extensions as our base field.
Let be a rational function field and define the field automorphism with . We call the rational difference field over .
Let be a rational function field (i.e., the are transcendental among each other over the field and let be the rational difference field over . Consider a -extension of with and for . Now consider the field of fractions . We also use the shortcut and write . Then is a -field extension of the difference field . It is also called the mixed -multibasic difference field over . If which is free of , then is called the -multibasic difference field over . Finally, if , then and is called a - or a basic difference field over .
Based on these ground fields we will define now our products. In the first sections we will restrict to the hypergeometric case.
Let and let be a rational difference field. Then the product expressions
from can be represented in an -extension as follows. Here is the complex unit which we also write as . Now take the -extension of with of order . The -monomial models with the shift-behavior Further, is also an -extension of with of order . The generator models with .
The product expressions
from with are represented in a -extension of the rational difference field with as follows.
Consider the -extension of with , and . In this ring, we can model polynomial expressions in and with the shift behavior and . Here, and are rephrased by and , respectively.
Constructing the -extension of with , and , we are able to model polynomial expressions in and with the shift behavior and by rephrasing and with and , respectively.
Introducing the -extension of with , and , one can model polynomial expressions in and with the shift behavior and by rephrasing and by and , respectively.
The hypergeometric product expressions
from can be represented in a -extension defined over the rational difference field in the following way. Take the -extension of with and . In this extension, one can model polynomial expressions in the product expression with the shift behavior and by rephrasing and by and . Finally, taking the -extension of with and , we can represent polynomial expressions in the product expression with the shift-behavior and by rephrasing and by and , respectively.
In order to solve Problem RPE within the next sections, we rely on a refined construction of -extensions. More precisely, we will represent our products in -extensions.
An -extension (- or -extension) of is called an -extension (- or -extension) if . Depending on the type of extension, we call an -/-/-monomial. Similarly, let be a field. Then we call a tower of -extensions of a -extension if .
We concentrate mainly on product extensions and skip the sum part that has been mentioned in the introduction. Still, we need to handle the very special case of the rational difference field with or the mixed -multibasic version. Thus it will be convenient to introduce also the field version of -extensions karr1981summation ; schneider2001symbolic .
Let be a difference field extension of with transcendental over and with . This extension is called a -extension333In karr1981summation also the more general case with is treated. In the following we restrict to the simpler case which is less technical but general enough to cover all problems that we observed in practical problem solving. if . In this case is also called a -monomial. is called a -extension of if it is either a - or a -extension. I.e., is transcendental over , and is a -monomial ( for some ) or is a -monomial ( for some ). is a -extension of if it is a tower of -extensions.
Note that there exist criteria which can assist in the task to check if during the construction the constants remain unchanged. The reader should see (schneider2016difference, , Proof , and ) for the proofs. For the field version, see also karr1981summation .
Let be a difference ring. Then the following statements hold.
Let be a -extension of with where . Then this is a -extension (i.e., ) iff there are no and with .
Let be an -extension of of order with where . Then this is an -extension (i.e., ) iff there are no and with . If it is an -extension, is a primitive th root of unity.
Let be a field and let be a difference field extension of with transcendental over and with . Then this is a -extension (i.e., ) if there is no with .
Concerning our base case difference fields (see Examples 2.1 and 2.2) the following remarks are relevant. The rational difference field is a -extension of by part (3) of Theorem 2.1 and using the fact that there is no with . Thus . Furthermore, the mixed -multibasic difference field with is a -field extension of . See Corollary 5.1 below. As a consequence, we have that .
We give further examples and non-examples of -extensions.
In Example 2.3, the -extension is an -extension of of order since there are no and with . However, the -extension is not an -extension of since with and , we have . In particular, we get .
The -extension of in Example 2.4(1) with is a -extension of as there are no and with . Similarly, the -extension in Example 2.4(2) with is also a -extension of since there does not exist a and a with . However, the -extension in part (3) of Example 2.4 is not a -extension of since with we have . In particular, .
Finally, in Example 2.5 the -extension is a -extension of with since there are no and with . But the -extension with is not a -extension of since with and we have . In particular, we get .
We remark that in karr1981summation and schneider2016difference algorithms have been developed that can carry out these checks if the already designed difference ring is built by properly chosen -extensions. In this article we are more ambitious. We will construct -extensions carefully such that they are automatically -extensions and such that the products under consideration can be rephrased within these extensions straightforwardly. In this regard, we will utilize the following lemma.
Let be a -extension of with . Then the -extension of with order is an -extension.
2.2 Ring of sequences
We will elaborate how -extensions can be embedded into the difference ring of sequences schneider2017summation ; compare also van2006galois . Precisely this feature will enable us to handle condition (2) of 1.1.
Let be a field containing as a subfield and let be the set of non-negative integers. We denote by the set of all sequences whose terms are in . With component-wise addition and multiplication, forms a commutative ring. The field can be naturally embedded into as a subring, by identifying with the constant sequence . Following the construction in (petkovvsek1996b, , Sec. ), we turn the shift operator
into a ring automorphism by introducing an equivalence relation on sequences in . Two sequences and are said to be equivalent if and only if there exists a natural number such that for all . The set of equivalence classes form a ring again with component-wise addition and multiplication which we will denote by . For simplicity, we denote the elements of (also called germs) by the usual sequence notation as above. Now it is obvious that is a ring automorphism. Therefore, forms a difference ring called the (difference) ring of sequences (over ).
Let and be two difference rings. We say that is a difference ring homomorphism between the difference rings and if is a ring homomorphism and for all , . If is injective then it is called a difference ring monomorphism or a difference ring embedding. In this case is a sub-difference ring of where and are the same up to renaming with respect to . If is a bijection, then it is a difference ring isomorphism and we say and are isomorphic.
Let be a difference ring with constant field . A difference ring homomorphism (resp. monomorphism) is called -homomorphism (resp. -monomorphism) if for all we have that .
The following lemma is the key tool to embed difference rings constructed by -extensions into the ring of sequences.
Let be a difference ring with constant field . Then:
The map is a -homomorphism if and only if there is a map with for all satisfying the following properties:
for all , there is a natural number such that