An identity for the sum of inverses of odd divisors of n in terms of the number of representations of n as a sum of r squares
aa r X i v : . [ m a t h . G M ] J u l An identity for the sum of inverses of odddivisors of n in terms of the number of rep-resentations of n as a sum of r squares Sumit Kumar Jha
Abstract.
Let X d | nd ≡ d denote the sum of inverses of odd divisors of a positive integer n , and let c r ( n ) be the number of representations of n as a sum of r squares whererepresentations with different orders and different signs are counted asdistinct. The aim is of this note is to prove the following interestingcombinatorial identity X d | nd ≡ d = 12 n X r =1 ( − n + r r nr ! c r ( n ) . Mathematics Subject Classification (2010).
Primary 11A05.
Keywords.
Sum of divisors; Sum of squares.
1. Main result
In the following, let X d | nd ≡ d denote the sum of inverses of odd divisors of a positive integer n . Definition 1.1. [2, formula 7.324] Let θ ( q ) be the following infinite product θ ( q ) := ∞ Y j =1 − q j q j = ∞ X n = −∞ ( − n q n where | q | <
1. S. K. Jha
Definition 1.2.
For any positive integer r define c r ( n ) by θ ( q ) r = ∞ X n =0 c r ( n )( − n q n where c r ( n ) be the number of representations of n as a sum of r squareswhere representations with different orders and different signs are counted asdistinct.Our aim is to derive the following identity. Theorem 1.
For all positive integers n we have X d | nd ≡ d = 12 n X r =1 ( − n + r r (cid:18) nr (cid:19) c r ( n ) . (1)We require following two lemmas for our proof. Lemma 1.
For all positive integers n we have2 X d | nd ≡ d = 1 n ! n X k =1 ( − k ( k − B n,k (cid:16) θ ′ (0) , θ ′′ (0) , . . . , θ ( n − k +1) (0) (cid:17) (2)where B n,k ≡ B n,k ( x , x , . . . , x n − k +1 ) are the partial Bell polynomials de-fined by [1, p. 206]B n,k ( x , x , . . . , x n − k +1 ) = X ≤ i ≤ n,ℓ i ∈ N P ni =1 iℓ i = n P ni =1 ℓ i = k n ! Q n − k +1 i =1 ℓ i ! n − k +1 Y i =1 (cid:16) x i i ! (cid:17) ℓ i . Proof.
It is easy to see thatlog( θ ( q )) = ∞ X j =1 log(1 − q j ) − ∞ X j =1 log(1 + q j )= − ∞ X j =1 ∞ X l =1 q lj l + ∞ X j ′ =1 ∞ X l ′ =1 q l ′ j ′ ( − l ′ l ′ = − ∞ X n =1 q n X d | n − ( − d d . Let f ( q ) = log q . Using Fa`a di Bruno’s formula [1, p. 134] we have d n dq n f ( θ ( q )) = n X k =1 f ( k ) ( θ ( q )) · B n,k (cid:16) θ ′ ( q ) , θ ′′ ( q ) , . . . , θ ( n − k +1) ( q ) (cid:17) . (3)Since f ( k ) ( q ) = ( − k − ( k − q k and θ (0) = 1, letting q → (cid:3) combinatorial identity for sum of divisors 3 Lemma 2.
We have for positive integers n, kB n,k (cid:16) θ ′ (0) , θ ′′ (0) , . . . , θ ( n − k +1) (0) (cid:17) = ( − n n ! k ! k X r =1 ( − k − r (cid:18) kr (cid:19) c r ( n ) (4) Proof.
We start with the generating function for the partial Bell polynomialsas follows ∞ X n = k B n,k (cid:16) θ ′ (0) , θ ′′ (0) , . . . , θ ( n − k +1) (0) (cid:17) q n n ! = 1 k ! ∞ X j =1 θ ( j ) (0) q j j ! k = 1 k ! ( θ ( q ) − k = 1 k ! k X r =0 ( − k − r (cid:18) kr (cid:19) θ ( q ) r = 1 k ! k X r =0 ( − k − r (cid:18) kr (cid:19) ∞ X n =0 ( − n c r ( n ) q n to conclude the equation (4). (cid:3) Proof of Theorem 1.
Combining equations (2) and (4) we have2 X d | nd ≡ d = ( − n n X k =1 k k X r =1 ( − r (cid:18) kr (cid:19) c r ( n )= ( − n n X r =1 ( − r c r ( n ) n X k = r k (cid:18) kr (cid:19) = ( − n n X r =1 ( − r r (cid:18) nr (cid:19) c r ( n ) . Now we can conclude our main result equation (1). (cid:3)
References [1] L. Comtet,
Advanced Combinatorics: The Art of Finite and Infinite Expan-sions , D. Reidel Publishing Co., Dordrecht, 1974.[2] N. J. Fine,
Basic Hypergeometric Series and Applications , American Mathe-matical Soc., 1988.Sumit Kumar JhaInternational Institute of Information TechnologyHyderabad-500 032, Indiae-mail:, American Mathe-matical Soc., 1988.Sumit Kumar JhaInternational Institute of Information TechnologyHyderabad-500 032, Indiae-mail: