The WZ method and zeta function identities Ira M. Gessel Department - PowerPoint PPT Presentation

The WZ method and zeta function identities Ira M. Gessel Department of Mathematics Brandeis University American Mathematical Society 2010 Fall Eastern Sectional Meeting Syracuse University October 3, 2010

There are several interesting formulas that give quickly converging series for generating functions for the zeta function, due to Almkvist, Borwein, Bradley, Granville, Koecher, Leshchiner, and Rivoal.

There are several interesting formulas that give quickly converging series for generating functions for the zeta function, due to Almkvist, Borwein, Bradley, Granville, Koecher, Leshchiner, and Rivoal. For example, Koecher’s formula is � n − 1 ∞ ∞ ( − 1 ) n + 1 1 − x 2 � 1 � � 2 ζ ( 2 k + 3 ) x 2 k = � � � 2 + . n 3 � 2 n 1 − x 2 / n 2 k 2 � n k = 0 n = 1 k = 1

Khodabakhsh and Tatiana Hessami Pilehrood used the WZ method of Wilf and Zeilberger to prove some of these formulas.

Khodabakhsh and Tatiana Hessami Pilehrood used the WZ method of Wilf and Zeilberger to prove some of these formulas. I will give a brief explanation of the WZ method and I will explain how the WZ proofs of these zeta function identities are closely related to classical hypergeometric series identities, and how this connection allows us to generalize them.

The WZ method We consider a grid graph with a weight function defined on every directed edge with the following property: ◮ For all vertices A and B , all paths from A to B have the same weight. (Path-invariance) 1 2 2 1 1 1 1 1 2 2 1 2 0 2 1 1 4

This property is equivalent to the existence of a potential function defined on the vertices: the weight of an edge is the difference in the potential of its endpoints: 6 8 3 4 2 2 1 1 1 1 1 3 2 5 7 1 2 2 1 0 2 2 1 1 1 4 5 6 0

Some observations ◮ To check that a weighted grid graph has the path invariance property, it is sufficient to check it on each rectangle: b a d a + b = c + d c

Some observations ◮ The weights of a directed edge and its reversal are negatives of each other.

Some observations ◮ We can add arbitrary other edges to the graph, and they will have uniquely determined weights that satisfy the path- invariance property. 1 2 2 1 1 1 1 1 2 2 1 2 0 2 1 1 4

In particular, from one path-invariant weighted grid graph, we can get many more by taking a different grid on the same set of points:

In particular, from one path-invariant weighted grid graph, we can get many more by taking a different grid on the same set of points:

In particular, from one path-invariant weighted grid graph, we can get many more by taking a different grid on the same set of points:

In particular, from one path-invariant weighted grid graph, we can get many more by taking a different grid on the same set of points:

The WZ method Suppose that we have a a path-invariant weighted grid graph, where the vertices are in Z × Z . Let f ( i , j ) be the weight on the edge from ( i , j ) to ( i + 1 , j ) and let g ( i , j ) be the weight on the edge from ( i , j ) to ( i , j + 1 ) : f ( i, j + 1) g ( i, j ) g ( i + 1 , j ) ( i, j ) f ( i, j )

The WZ method Suppose that we have a a path-invariant weighted grid graph, where the vertices are in Z × Z . Let f ( i , j ) be the weight on the edge from ( i , j ) to ( i + 1 , j ) and let g ( i , j ) be the weight on the edge from ( i , j ) to ( i , j + 1 ) : f ( i, j + 1) g ( i, j ) g ( i + 1 , j ) ( i, j ) f ( i, j ) The path invariance property is f ( i , j ) + g ( i + 1 , j ) = g ( i , j ) + f ( i , j + 1 )

The WZ method Suppose that we have a a path-invariant weighted grid graph, where the vertices are in Z × Z . Let f ( i , j ) be the weight on the edge from ( i , j ) to ( i + 1 , j ) and let g ( i , j ) be the weight on the edge from ( i , j ) to ( i , j + 1 ) : f ( i, j + 1) g ( i, j ) g ( i + 1 , j ) ( i, j ) f ( i, j ) The path invariance property is f ( i , j ) + g ( i + 1 , j ) = g ( i , j ) + f ( i , j + 1 ) Note: Traditionally the order of the parameters is switched.

A pair of functions ( f , g ) satisfying this identity is called a WZ-pair . For our purposes, we want to assume that f and g have a particular form: we want each to be of the form Γ( a 1 i + b 1 j + u 1 ) · · · Γ( a k i + b k j + u k ) Γ( c 1 i + d 1 j + v 1 ) · · · Γ( c l i + d l j + v l ) z i w j where the a n , b n , c n , and d n are integers, and the u n , v n , z , and w are complex numbers.

A pair of functions ( f , g ) satisfying this identity is called a WZ-pair . For our purposes, we want to assume that f and g have a particular form: we want each to be of the form Γ( a 1 i + b 1 j + u 1 ) · · · Γ( a k i + b k j + u k ) Γ( c 1 i + d 1 j + v 1 ) · · · Γ( c l i + d l j + v l ) z i w j where the a n , b n , c n , and d n are integers, and the u n , v n , z , and w are complex numbers. This implies that f ( i + 1 , j ) / f ( i , j ) , f ( i , j + 1 ) / f ( i , j ) , and f ( i , j ) / g ( i , j ) are rational functions.

A pair of functions ( f , g ) satisfying this identity is called a WZ-pair . For our purposes, we want to assume that f and g have a particular form: we want each to be of the form Γ( a 1 i + b 1 j + u 1 ) · · · Γ( a k i + b k j + u k ) Γ( c 1 i + d 1 j + v 1 ) · · · Γ( c l i + d l j + v l ) z i w j where the a n , b n , c n , and d n are integers, and the u n , v n , z , and w are complex numbers. This implies that f ( i + 1 , j ) / f ( i , j ) , f ( i , j + 1 ) / f ( i , j ) , and f ( i , j ) / g ( i , j ) are rational functions. Note: We will want to allow i and j to be complex numbers.

We illustrate our approach with the simplest interesting WZ pair, which is associated with the binomial theorem. One form of this WZ pair is Γ( i + j ) � i + j − 1 � Γ( i + 1 )Γ( j ) x i ( 1 − x ) j = x i ( 1 − x ) j f ( i , j ) = i Γ( i + j ) � i + j − 1 � Γ( i )Γ( j + 1 ) x i ( 1 − x ) j = − x i ( 1 − x ) j g ( i , j ) = − i − 1 Let’s first see the connection between this WZ pair and the binomial theorem.

We sum along the two paths P 1 and P 2 : ( m, n ) (0 , n ) P 2 P 1 (0 , 0) ( m, 0) Summing along P 1 gives m − 1 n − 1 n − 1 � m + j − 1 � � � � x m ( 1 − x ) j . f ( i , 0 ) + g ( m , j ) = 1 − i − 1 i = 0 j = 0 j = 0 and summing along P 2 gives n − 1 m − 1 m − 1 � i + n − 1 � � � � x i ( 1 − x ) n . g ( 0 , j ) + f ( i , n ) = 0 + i j = 0 i = 0 i = 0

So we have the identity n − 1 m − 1 � m + j − 1 � � i + n − 1 � x m ( 1 − x ) j = � � x i ( 1 − x ) n . 1 − i − 1 i j = 0 i = 0 Taking the limit as m → ∞ we get the binomial theorem in the form ∞ � i + n − 1 � � x i ( 1 − x ) n . 1 = i i = 0

So we have the identity n − 1 m − 1 � m + j − 1 � � i + n − 1 � x m ( 1 − x ) j = � � x i ( 1 − x ) n . 1 − i − 1 i j = 0 i = 0 Taking the limit as m → ∞ we get the binomial theorem in the form ∞ � i + n − 1 � � x i ( 1 − x ) n . 1 = i i = 0 Conversely, starting with the binomial theorem in this form, it’s easy to find f and g , using Gosper’s algorithm.

We can use this WZ pair to obtain identities for the sum ∞ α � α + i x i , L ( x , α ) = i = 0 which can be expressed in terms of the Lerch transcendent as α Φ( x , 1 , α ) .

We can use this WZ pair to obtain identities for the sum ∞ α � α + i x i , L ( x , α ) = i = 0 which can be expressed in terms of the Lerch transcendent as α Φ( x , 1 , α ) . Note that L ( x , 1 ) = − x − 1 log ( 1 − x ) and L ( x , 1 / 2 ) = x − 1 / 2 tanh − 1 √ x .

The sum L ( x , α ) is a generating function for the polygorithm x i ∞ � Li s ( x ) = i s i = 1 which reduces to the zeta function for x = 1.

The sum L ( x , α ) is a generating function for the polygorithm x i ∞ � Li s ( x ) = i s i = 1 which reduces to the zeta function for x = 1. We have ∞ ∞ x i α 1 α + i x i = 1 + α � � L ( x , α ) = i 1 + α/ i i = 0 i = 1 ∞ ∞ x i � ( − 1 ) m α m + 1 � = 1 + i m + 1 m = 0 i = 1 ∞ = 1 + 1 ( − 1 ) m − 1 α m Li m ( x ) . � x m = 1

Let’s look at Γ( i + j ) Γ( i + 1 )Γ( j ) x i ( 1 − x ) j . f ( i , j ) = We want to modify f and g so that � ∞ i = 0 f ( i , 0 ) becomes i = 0 α x i / ( α + i ) . something like � ∞

Let’s look at Γ( i + j ) Γ( i + 1 )Γ( j ) x i ( 1 − x ) j . f ( i , j ) = We want to modify f and g so that � ∞ i = 0 f ( i , 0 ) becomes i = 0 α x i / ( α + i ) . If we replace i with α + i in something like � ∞ f ( i , j ) , then we get Γ( α + i + 1 ) x i · x α ( 1 − x ) j f ( α + i , j ) = Γ( α + i + j ) Γ( j )

Let’s look at Γ( i + j ) Γ( i + 1 )Γ( j ) x i ( 1 − x ) j . f ( i , j ) = We want to modify f and g so that � ∞ i = 0 f ( i , 0 ) becomes i = 0 α x i / ( α + i ) . If we replace i with α + i in something like � ∞ f ( i , j ) , then we get Γ( α + i + 1 ) x i · x α ( 1 − x ) j f ( α + i , j ) = Γ( α + i + j ) Γ( j ) Now if we set j = 0, we get x α α α + i x i · f ( α + i , 0 ) = α Γ( 0 ) which is just what we want except for a constant factor