Closed forms for generating series, and finite summation analogues - PowerPoint PPT Presentation

Closed forms for generating series, and finite summation analogues modulo a prime Sandro Mattarei (joint work with Roberto Tauraso) School of Mathematics and Physics University of Lincoln (UK) Gargnano, October 2017 S. Mattarei Finite analogues of generating functions Gargnano, October 2017 1 / 14

A famous series and a finite analogue k =1 1 / k 3 is irrational Apéry’s celebrated proof (1979) that ζ (3) = � ∞ relied on the fast converging series ( − 1) k ∞ � = − 2 � 5 ζ (3) . k 3 � 2 k k =1 k S. Mattarei Finite analogues of generating functions Gargnano, October 2017 2 / 14

A famous series and a finite analogue k =1 1 / k 3 is irrational Apéry’s celebrated proof (1979) that ζ (3) = � ∞ relied on the fast converging series ( − 1) k ∞ � = − 2 � 5 ζ (3) . k 3 � 2 k k =1 k Tauraso (2010): for any prime p > 5 we have p − 1 ( − 1) k � ≡ 2 H p − 1 � (mod p 3 ) , k 3 � 2 k p 2 5 k =1 k where H k = � p − 1 k =1 1 / k denote the harmonic numbers. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 2 / 14

A famous series and a finite analogue k =1 1 / k 3 is irrational Apéry’s celebrated proof (1979) that ζ (3) = � ∞ relied on the fast converging series ( − 1) k ∞ � = − 2 � 5 ζ (3) . k 3 � 2 k k =1 k Tauraso (2010): for any prime p > 5 we have p − 1 ( − 1) k � ≡ 2 H p − 1 � (mod p 3 ) , k 3 � 2 k p 2 5 k =1 k where H k = � p − 1 k =1 1 / k denote the harmonic numbers. � for ( p − 1) / 2 ≤ k < p , but because of � 2 k Note that p divides k cancellation the LHS turns out to be p -integral. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 2 / 14

A famous series and a finite analogue k =1 1 / k 3 is irrational Apéry’s celebrated proof (1979) that ζ (3) = � ∞ relied on the fast converging series ( − 1) k ∞ � = − 2 � 5 ζ (3) . k 3 � 2 k k =1 k Tauraso (2010): for any prime p > 5 we have p − 1 ( − 1) k � ≡ 2 H p − 1 � (mod p 3 ) , k 3 � 2 k p 2 5 k =1 k where H k = � p − 1 k =1 1 / k denote the harmonic numbers. � for ( p − 1) / 2 ≤ k < p , but because of � 2 k Note that p divides k cancellation the LHS turns out to be p -integral. Note also that H p − 1 ≡ 0 (mod p 2 ) by Wolstenholme’s theorem. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 2 / 14

A famous series and a finite analogue k =1 1 / k 3 is irrational Apéry’s celebrated proof (1979) that ζ (3) = � ∞ relied on the fast converging series ( − 1) k ∞ � = − 2 � 5 ζ (3) . k 3 � 2 k k =1 k Tauraso (2010): for any prime p > 5 we have p − 1 ( − 1) k � ≡ 2 H p − 1 � (mod p 3 ) , k 3 � 2 k p 2 5 k =1 k where H k = � p − 1 k =1 1 / k denote the harmonic numbers. � for ( p − 1) / 2 ≤ k < p , but because of � 2 k Note that p divides k cancellation the LHS turns out to be p -integral. Note also that H p − 1 ≡ 0 (mod p 2 ) by Wolstenholme’s theorem. There is a clear similarity in the coefficient 2 / 5, but the rest of the analogy is more mysterious. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 2 / 14

Another famous series � 3 � ∞ − 1 / 2 = 2 � (4 k + 1) (Ramanujan) k π k =0 S. Mattarei Finite analogues of generating functions Gargnano, October 2017 3 / 14

Another famous series � 3 � ∞ − 1 / 2 = 2 � (4 k + 1) (Ramanujan) k π k =0 � � � � 2 k − 1 / 2 = ( − 4) k Note that . k k S. Mattarei Finite analogues of generating functions Gargnano, October 2017 3 / 14

Another famous series � 3 � ∞ − 1 / 2 = 2 � (4 k + 1) (Ramanujan) k π k =0 � � � � 2 k − 1 / 2 = ( − 4) k Note that . k k Van Hamme (1996) conjectured, and Mortenson (2008) proved � 3 p − 1 � − 1 / 2 � − 1 � p (mod p 3 ) � (4 k + 1) ≡ k p k =0 � � − 1 for p > 2, where is a Legendre symbol. p S. Mattarei Finite analogues of generating functions Gargnano, October 2017 3 / 14

Another famous series � 3 � ∞ − 1 / 2 = 2 � (4 k + 1) (Ramanujan) k π k =0 � � � � 2 k − 1 / 2 = ( − 4) k Note that . k k Van Hamme (1996) conjectured, and Mortenson (2008) proved � 3 p − 1 � − 1 / 2 � − 1 � p (mod p 3 ) � (4 k + 1) ≡ k p k =0 � � − 1 for p > 2, where is a Legendre symbol. p One can only see the analogy after re-writing the right-hand sides in terms of values of the gamma function, and of the p -adic gamma function, each evaluated at 1 / 2 in this case. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 3 / 14

And a less famous series Some numerical series are specializations of power series in an indeterminate which admit a closed form . Not so for the series used by Apéry, or that of Ramanujan, but for example, � = π 2 ∞ 1 � k 2 � 2 k 18 k =1 k S. Mattarei Finite analogues of generating functions Gargnano, October 2017 4 / 14

And a less famous series Some numerical series are specializations of power series in an indeterminate which admit a closed form . Not so for the series used by Apéry, or that of Ramanujan, but for example, � = π 2 ∞ 1 � k 2 � 2 k 18 k =1 k comes from � √ x �� 2 x k ∞ � � � = 2 arcsin k 2 � 2 k 2 k =1 k S. Mattarei Finite analogues of generating functions Gargnano, October 2017 4 / 14

And a less famous series Some numerical series are specializations of power series in an indeterminate which admit a closed form . Not so for the series used by Apéry, or that of Ramanujan, but for example, � = π 2 ∞ 1 � k 2 � 2 k 18 k =1 k comes from � √ x �� 2 x k ∞ � � � = 2 arcsin k 2 � 2 k 2 k =1 k Mattarei/Tauraso (2013): This has a finite analogue mod p 2 : p − 1 � ≡ 2 − α p − α − p − x p x k � (mod p 2 ) , p k 2 � 2 k 2 p k =1 k where α 2 + ( x − 2) α + 1 = 0, and one more complicated mod p 3 . S. Mattarei Finite analogues of generating functions Gargnano, October 2017 4 / 14

And a less famous series Some numerical series are specializations of power series in an indeterminate which admit a closed form . Not so for the series used by Apéry, or that of Ramanujan, but for example, � = π 2 ∞ 1 � k 2 � 2 k 18 k =1 k comes from � √ x �� 2 x k ∞ � � � = 2 arcsin k 2 � 2 k 2 k =1 k Mattarei/Tauraso (2013): This has a finite analogue mod p 2 : p − 1 � ≡ 2 − α p − α − p − x p x k � (mod p 2 ) , p k 2 � 2 k 2 p k =1 k where α 2 + ( x − 2) α + 1 = 0, and one more complicated mod p 3 . Hard to see a similarity in this case. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 4 / 14

Two tricks in action on a simple example We look at a class of power series having a closed form, which have a finite analogue mod p , also with a closed form. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 5 / 14

Two tricks in action on a simple example We look at a class of power series having a closed form, which have a finite analogue mod p , also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 5 / 14

Two tricks in action on a simple example We look at a class of power series having a closed form, which have a finite analogue mod p , also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p . Shorten ‘ ≡ (mod p )’ to ‘ ≡ p ’. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 5 / 14

Two tricks in action on a simple example We look at a class of power series having a closed form, which have a finite analogue mod p , also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p . Shorten ‘ ≡ (mod p )’ to ‘ ≡ p ’. Conditions such as p > 2 will be omitted for simplicity. S. Mattarei Finite analogues of generating functions Gargnano, October 2017 5 / 14

Two tricks in action on a simple example We look at a class of power series having a closed form, which have a finite analogue mod p , also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p . Shorten ‘ ≡ (mod p )’ to ‘ ≡ p ’. Conditions such as p > 2 will be omitted for simplicity. Here is a simple example. q − 1 � � � � ∞ 2 k 1 2 k x k = x k ≡ p (1 − 4 x ) ( q − 1) / 2 � � √ ⇒ k k 1 − 4 x k =0 k =0 S. Mattarei Finite analogues of generating functions Gargnano, October 2017 5 / 14