[seqfan] Re: Sums of Binomials

[Vladimir Shevelev]

Dear SeqFans,

Yesterday I have done  the text
of my paper 
http://arxiv.org/abs/1706.01454
more clear (see,for example, the 
additional definition (10) and the
explanation that it is quite natural)
In my previous message I used N
instead of n and n instead of m since
it is more suitable for OEIS. Now
I continue in this style.
For N=4 we have in OEIS the full 
set of sequences for {H_i},i=1..4,
but in A038505 there is an error:
a(1)=0 is missed. Let us discuss
A000749. Here omega=i=sqrt(-1)
and according Theorem 1 in my paper
H_4=Sum{t>=0}C(n,4t+3)=
(1/4)Sum{j=1..4}(i^j+1)^n*i^(-3j)=
1/4(1+i)^n*i^(-3)+0+(1-i)^n*i^(-9)
+2^n)=1/4((1+i)^n*i-(1-i)^n*i+2^n).
This satisfies the recursion in the name,
since its characteristic equation has
the roots 2,1+i,1-i,
but if n=0 we have 1/4. Is it error?
No! The zero above appears only for
n>0, but for n=0 we have 0^0*i^(-6)
=-1 and we now have instead of  1/4
the required 0.  

Best regards,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 11 June 2017 13:44
To: seqfan at list.seqfan.eu
Subject: [seqfan] Sums of Binomials

Dear Seqfans,

For a fixed N>=2, consider sums binomials
H_i(n,N)=Sum{t>=0} C(n, Nt+i-1), i=1..N, n>=0,
K_i(n,N)=Sum{t>=0}(-1)^t C(n, Nt+i-1), i=1..N, n>=0.

I proved in
http://arxiv.org/abs/1706.01454
that the set {H_i(n,N)} is a difference analog of
the hyperbolic functions {h_i(x,N)} of order N which are,
by the original definition,
h_i(x,N)=(1/N)*Sum{1<=t<=N} omega^((1-i)t)*
exp((omega^t)x), i=1..N, where omega=exp(2(Pi)*j/N),
j=sqrt(-1),  while {K_i(n,N)}  is a difference analog of
the trigonometric functions {k_i(x,N)} of order N  which are,
again by the original definition, k_i(x,N)=
Sum{t>=0}((-1)^t)x^(Nt+i-1)/(Nt+i-1)!, i=1..N.

Besides, I found the addition formulas for H_i(n,N) and
K_i(n,N). For example, in the case N=3, we have
 H_1(n+m)=H_1(n)H_1(m)+H_2(n)H_3(m)+H_3(n)H_2(m),
 K_1(n+m)=K_1(n)K_1(m) - K_2(n)K_3(m) - K_3(n)K_2(m).

I wanted to find these sequences in OEIS. For N=2, they
are {A146559, A009545}; for N=3,
{H_1(n),H_2(n),H_3(n)}={A024493,A131708,A024495}
(note that A024494 is less suitable than A131708);
{K_1(n),K_2(n),K_3(n)}=A057681,A057682, ?).
A close to K_3(n) is A057083, but it is not suitable in view
 of its offset and it misses the two first 0's.
For N=4, {H_1(n),H_2(n),H_3(n),H_4(n)}={A038503,A038504,
A038505,A000749, absent};
{K_1,K_2,K_3,K_4}={A099586, absent,absent,absent}.
For N=5, {H_1,H_2,H_3,H_4,H_5}={A139398,A133476,
A139714,A139748,A139761}, but all K_i,1,...,5 are absent.
Can I make up for the absent sequence (up to N=5) and give
a suitable option for K_3 for N=3?

Best regards,
Vladimir




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