PROPERTIES OF RATIONAL ARITHMETIC FUNCTIONS

PROPERTIES OF RATIONAL ARITHMETIC FUNCTIONS

VICHIAN LAOHAKOSOL AND NITTIYA PABHAPOTE Received 13 January 2005 and in revised form 20 September 2005

Rational arithmetic functions are arithmetic functions of the formg1∗ ··· ∗gr∗h⁻₁¹∗

··· ∗h⁻_s¹, wheregi,hjare completely multiplicative functions and∗denotes the Dirich- let convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type identities are derived; and finally, properties of the Kesava Menon norm of such functions are proved.

1. Introduction

By anarithmetic functionwe mean a complex-valued function whose domain is the set of positive integersN. We define the addition and the Dirichlet convolution of two arith- metic functions f andg, respectively, by

(f+g)(n)=f(n) +g(n), (f∗g)(n)=

i j=n

f(i)g(j). (1.1) It is well known (see, e.g., [1,5,13,19,21]) that the set (Ꮽ, +,∗) of all arithmetic func- tions is a unique factorization domain with the arithmetic function

I(n)=





1 ifn=1,

0 otherwise, (1.2)

being its convolution identity.

A nonzero arithmetic function f ∈Ꮽis calledmultiplicative, denoted by f ∈ᏹ, if f(mn)= f(m)f(n) whenever (m,n)=1. It is calledcompletely multiplicative, denoted by f ∈Ꮿ, if f(mn)= f(m)f(n) for allm,n∈N.

For nonnegative integersr,s by an (r,s)-rational arithmetic function f, denoted by f ∈Ꮿ(r,s), we mean an arithmetic function which can be written as

f =g1∗ ··· ∗gr∗h⁻1¹∗ ··· ∗h⁻_s¹, (1.3)

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:24 (2005) 3997–4017 DOI:10.1155/IJMMS.2005.3997

where eachg_i,h_j∈Ꮿ. Such functions were first studied by Vaidyanathaswamy [23] in 1931, and later by several authors; see, for example, [4,6,7,9,10,13,16,18,20]. Two important classes of rational functions areᏯ(1, 1) whose elements are known astotients, andᏯ(2, 0) whose elements are the so-calledspecially multiplicative functions. Character- izations of these two classes can be found in [7,10], respectively.

The present work deals with four aspects of rational arithmetic functions. In the next section, some characterizations of these functions are derived and are then used in the next sections to investigate whether two types of identities, the Busche-Ramanujan iden- tity and the binomial identity, which are known to hold for totients and/or specially mul- tiplicative functions, continue to hold for general rational arithmetic functions. In the last section, the Kesava Menon norm of such functions is studied.

We will find it helpful to make use of two important concepts which we now recall. For f ∈Ꮽ, f(1)∈R⁺, the Rearick logarithm of f (see [11,14,15]), denoted by Logf ∈Ꮽ, is defined via

(Logf)(1)=logf(1), (Logf)(n)= 1

logn

d|n

f(d)f⁻¹ n

d

logd= 1 logn

df∗f⁻¹(n) (n >1), (1.4)

wheredf(n)= f(n) logndenotes the log derivation of f. The Hsu’s generalized M¨obius function (see [2])µ_r,r∈R, is defined as

µr(n)=

p|n

r νp(n)

(−1)^νp(n), (1.5)

whereνp(n) is the highest power of the primepdividingn. It is known (see [8,12]) that for f ∈ᏹ,

f ∈Ꮿ=⇒ f^r=µ₋rf, (1.6)

and the converse holds under additional hypotheses.

2. Characterizations

In this section,randswill generally denote nonnegative integers. Should either of them be zero, the sum and/or any other expressions connected with them are taken to be zero.

Theorem2.1. Let r,sbe nonnegative integers and f ∈ᏹ. Then, f ∈Ꮿ(r,s)⇔for each primepand eachα∈N, there exist complex numbersa1(p),. . .,ar(p),b1(p),. . .,bs(p)such that

(Logf)(n)=





 1 α

a1(p)^α+···+ar(p)^α−b1(p)^α− ··· −bs(p)^α ifn=p^α,

0 otherwise.

(2.1)

Proof.

f ∈Ꮿ(r,s)⇐⇒f =g1∗ ··· ∗gr∗h⁻₁¹∗ ··· ∗h⁻_s¹ gi,hj∈Ꮿ

⇐⇒Logf =Logg1+···+ Loggr−Logh1− ··· −Loghs. (2.2) The result now follows immediately from Carroll’s theorem [3] which states that for F∈ᏹ,

F∈Ꮿ⇐⇒(LogF)(n)=





 1

αF(p)^α ifn=p^α,

0 otherwise.

(2.3) Taking a1(p)^α= f(p^α+1)/ f(p), b1(p)=b(p) in Theorem 2.1, we get the following corollary.

Corollary2.2. Let f ∈ᏹ, withf(p)=0for each primep. Thenf ∈Ꮿ(1, 1)⇐⇒for each primepand eachα∈N, there is a complex numberb(p)such that

(Logf)(n)=







 1 α

fp^α+1

f(p) ⁻b(p)^α

ifn=p^α,

0 otherwise.

(2.4)

Theorem2.3. Letr,s be nonnegative integers and f ∈ᏹ. Then f ∈Ꮿ(r,s)⇔for each primepand eachα∈N, there exist complex numbersa1(p),. . .,ar(p),b1(p),. . .,bs(p)such that for allα_≥s,

fp^α= s k=0

Gα−kHk, (2.5)

where

Gα−k=

j1+···+jr=α−k

a1(p)^j1···ar(p)^jr, G0=1, H_k=(−1)^k

1_≤i1<i2<···<ik≤s

b_i1(p)···b_ik(p), H0=1. (2.6) Proof.

f ∈Ꮿ(r,s)⇐⇒f =g1∗ ··· ∗gr∗h⁻₁¹∗ ··· ∗h⁻_s¹ gi,hj∈Ꮿ

⇐⇒fp^α=

j1+···+jr+k1+···+ks=α

g1(p)^j1···g_r(p)^jrh⁻₁¹p^k1···h⁻_s¹p^ks. (2.7) The result now follows by grouping terms on the right-hand side and usingh⁻¹(p^k)=0

fork≥2.

A few known characterizations of two particular classes of functions, namely, those inᏯ(1, 1), that is, totients (see [7]), and those inᏯ(2, 0), that is, specially multiplicative functions (see [13, Theorem 1.12]), are immediate consequences ofTheorem 2.3, which we record in the following corollary together with a characterizing property ofᏯ(1,s) to be used later.

Corollary2.4. Let f ∈ᏹ. Then the following hold.

(i) f ∈Ꮿ(1, 1)⇔for each primepand eachα∈N, there exists a complex numbera(p) such that

fp^α=a(p)^α−1f(p). (2.8)

(ii) f ∈Ꮿ(2, 0)⇔for each primepand eachα(≥2)∈N,

fp^α+1= f(p)fp^α+fp^α−1fp²−f(p)². (2.9) (iii) f ∈Ꮿ(1,s)⇔for each primepand eachα∈N, there exist complex numbersa(p),

b1(p),. . .,bs(p)such that for allα≥s, fp^α=

s k=0

g(p)^α−kH_k, (2.10)

where

Hk=(−1)^k

1≤i1<i2<···<ik≤s

bi1(p)···bik(p), H0=1. (2.11) Simplified characterizations for rational arithmetic functions belonging to the classes whereris 0 can similarly be obtained as in the next corollaries.

Corollary2.5. Letsbe a nonnegative integer and f ∈ᏹ. Then f ∈Ꮿ(0,s)⇔for each primep, f(p^α)=0for allα > s.

Proof.

f ∈Ꮿ(0,s)⇐⇒f =h⁻₁¹∗ ··· ∗h⁻_s¹ hi∈Ꮿ

⇐⇒fp^α=

i1+_···+is=α

h⁻₁¹pⁱ¹···h⁻_s¹p^is. (2.12) The result now follows by noting that forh∈Ꮿ, we haveh⁻¹(p)= −h(p),h⁻¹(pⁱ)=0 fori≥2, and that thescomplex numbersh1(p),. . .,hs(p) are uniquely determined by the svalues f(p),. . .,f(p^s), which are generally arbitrary. In fact, by elementary symmetric functions, we note thath1(p),. . .,h_s(p) are just all thesroots of

X^s+f(p)X^s−1+···+fp^s−1X+fp^s=0. (2.13) This indeed renders their existence, which was stated in the result ofTheorem 2.3, to be

redundant.

Invoking upon the fact that f ∈Ꮿ(r, 0)⇔ f⁻¹∈Ꮿ(0,r), we easily deduce our next result which appears as [13, Problem 1.16, page 48].

Corollary2.6. Letrbe a nonnegative integer and f ∈ᏹ. Then

f ∈Ꮿ(r, 0)⇐⇒for each primep, f⁻¹p^α=0 ∀α > r. (2.14) Corollary2.7. Letrbe a nonnegative integer and f ∈ᏹ. Then f ∈Ꮿ(r, 0)⇔for each primep, and for allα_≥r,

fp^α+1= −

fp^αf⁻¹(p) + fp^α−1f⁻¹p²+···+fp^α−r+1f⁻¹p^r. (2.15) Proof. This follows by expanding f ∗f⁻¹=I at the prime powers p^αand applying the

result ofCorollary 2.6.

Recall that totients are elements ofᏯ(1, 1). It seems natural to further characterize a particular class ofᏯ(r,s), called here (r,s)-totients, defined by

f =g^r∗h^−s, g,h∈Ꮿ. (2.16)

Theorem2.8. Letr,sbe nonnegative integers, f ∈ᏹ. Thenf is an(r,s)-totient⇔for each primepand eachα(>2)∈N, there are complex numbersa(p),b(p)such that

fp^α=(−1)^α α i=0

−r α−i

s i

a(p)^α−ib(p)ⁱ. (2.17) Proof. Using the definition and properties of Hsu’s generalized M¨obius function men- tioned inSection 1, we have

f is an (r,s)-totient ⇐⇒f =g^r∗h^−s=µ_−rg∗µ_sh

⇐⇒fp^α= α i=0

µ₋rgp^α−iµshpⁱ. (2.18)

Takinga(p)=g(p),b(p)=h(p), the result follows.

Another important characterization ofᏯ(r,s) involving recurrence is due to Rutkowski [18] which states that f =g1∗ ··· ∗gr∗h⁻₁¹∗ ··· ∗h⁻_s¹∈Ꮿ(r,s)⇔for each primep and eachα∈N, there exist complex numbersc1(p),. . .,cr(p) such that

fp^α=c1(p)fp^α−1+···+cr(p)fp^α−r (α > s), (2.19) where

c1(p)= r i=1

g_i(p), c2(p)= −

1_≤i1<i2≤r

g_i1(p)g_i2(p),. . .,c_r(p)=(−1)^r+1g1(p)···g_r(p).

(2.20) We will have occasion to use Rutkowski’s result later.

3. Busche-Ramanujan-type identities

It is well known (see, e.g., [21, page 62], [7,10], or [13]) that

f ∈Ꮿ(2, 0)⇐⇒there existsB∈Ꮿsuch that for allm,n∈N, we have f(m)f(n)=

d|(m,n)

f mn

d²

B(d) (3.1)

⇐⇒there existsF∈ᏹsuch that for allm,n∈N, we have f(mn)=

d|(m,n)

f m

d

f n

d

F(d), (3.2)

and that

f ∈Ꮿ(1, 1)⇐⇒there existsh∈Ꮿsuch that for allm,n∈N, we have f(m)f(n)=

d|(m,n)

f mn

d

h(d)µ(d) (3.3)

⇐⇒there existsF∈ᏹsuch that for allm,n∈N, we have f(mn)=

d|(m,n)

f m

d

f n

d

F(d), (3.4)

whenever the greatest common unitary divisor (m,n)_u=1, f(p²)= f(p)²+F(p), and f(p)=0 for all primesp. For the notion of unitary divisor, see [21, page 9].

Identities (3.1) and (3.2) are known as Busche-Ramanujan identities, while (3.4) is called the restricted Busche-Ramanujan identity because of the restrictions onm,n. In this section, we ask whether similar identities hold for functions in generalᏯ(r,s). An earlier affirmative answer to a particular case of this problem appears in [9, Theorem 4.2]

which in our terminology states that for f =g1∗g2∗h⁻¹∈Ꮿ(2, 1), we have f(mn)=

d|(m,n)

g1∗g2

m d

f n

d

µ(d)g1(d)g2(d), (3.5)

wheneverγ(m)|γ(n), whereγ(m) denotesthe product of all distinct primes factors ofm.

We will show that there are similar Busche-Ramanujan-type identities for functions in the classesᏯ(r,s) withr=1, 2, but are possible forr≥3 with rather artificial flavor. As to be expected, the identities are of restricted form, that is, hold with conditions onm,n.

Definition 3.1. Let s be a nonnegative integer. A pair (m,n)∈N×N is said to be s- excessive if for each primepdividing (m,n), eitherνp(m)≥νp(n) +sorνp(n)≥νp(m) + s, whereνp(m) denotes the highest power ofpappearing inm.

Note that the 0-excessive pairs are trivially all pairs of natural numbers, while the 1- excessive pairs (m,n) correspond exactly to those with the greatest common unitary divi- sor (m,n)_u=1.

Theorem3.2. Letsbe a nonnegative integer and f =g∗h⁻₁¹∗ ··· ∗h⁻_s¹∈Ꮿ(1,s). For each primep, ifg(p)=0, and^s_k=0g(p)^s−kH_k=0, whereH_k=(−1)^k_{1≤i1<i2<···<ik≤s}h_i1(p)

···hik(p),H0=1, then there existsF∈ᏹsuch that f(mn)=

d|(m,n)

f m

d

f n

d

F(d), (3.6)

for eachs-excessive pair(m,n).

Proof. Since f ∈ᏹ, the identity holds for allm,nwith (m,n)=1. It thus remains to prove this identity when (m,n)>1. For such s-excessive pair (m,n), let their prime factorizations be

m=p1^a1···p_u^auq^c11¹···q^c_1v^v, n=p^b1¹···p^b_u^uq^d21¹···q_2w^dw, (3.7) wherep_i,q_1j,q_2kare distinct primes;a_i,b_j,c_k,d_lare positive integers. By multiplicativity, we can write

f(mn)=Q

u i=1

fp^a_i^i+bi, (3.8)

where

Q=fq^c₁₁¹···fq^c_1v^vfq^d₂₁¹···fq^d_2w^w. (3.9) The right-hand side of the identity becomes

d|(m,n)

f m

d

f n

d

F(d)=Q

u i=1

min(ai,bi) j=0

fp^a_i^i−jfp^b_i^i−jFp_i^j. (3.10) Assuming without loss of generality thatνp(m)≥νp(n) +s, that is,a≥b+s, the identity will be established if we can findF∈ᏹsatisfying

fp^a+b= b j=0

fp^a−jfp^b−jFp^j, (3.11) for each primep. It suffices to exhibitF(p^j), the values ofFat prime powers, independent ofaandb, such that

f_a+b= b j=0

f_a−jf_b−jF_j, (3.12)

where, for short, we put f(pⁱ)= f_i,F(p^j)=F_j. Substitutingb=1 into (3.12), we have fa+1=faf1+fa−1F1 (a≥s+ 1). (3.13)

Replacing f_a+1, f_a, f_a−1,f1usingCorollary 2.4(iii), we have s

k=0

g(p)^a+1−kH_k=

g(p)− s i=1

h_i(p) _s

k=0

g(p)^a−kH_k+F1

s k=0

g(p)^a−1−kH_k, (3.14)

yielding F1 =g(p)^s_i=1hi(p), which is independent of a, provided that g(p) and _s

k=0g(p)^a−1−kHkare nonzero. Substitutingb=2 into (3.12), we get

f_a+2=f_af2+f_a−1f1F1+f_a−2F2 (a≥s+ 2). (3.15) Replacing fa+2, fa, fa−1,f2, f1, usingCorollary 2.4(iii) and the value ofF1, we find that

F2=g(p)² _s

i=1

hi(p) 2

−

1≤i1<i2≤s

hi1(p)hi2(p)

, (3.16)

independent ofa. In general, for fixed j, fromCorollary 2.4(iii), witha−j≥s, we have f_a+j=g^2jf_a−j, f_a+j−1=g^2j−1f_a−j,. . ., f_a−j+1=g f_a−j. (3.17) Substituting these and the previous values ofF_i(i < j) into (3.12), and dividing by f_a−j, we uniquely determineFj independent ofa. Note that the division by fa−jis legitimate because fromg(p), fs=_s

k=0g^s−kHkbeing nonzero, we immediately infer that fa=0 for

alla≥s.

Theorem3.3. Lets∈N. If f =g1∗g2∗h⁻1¹∗ ··· ∗h⁻_s¹∈Ꮿ(2,s), then f(mn)=

d|(m,n)

g1∗g2

m d

f n

d

µ(d)g1g2

(d), (3.18)

for each(s−1)-excessive pair(m,n)withγ(m)|γ(n).

Proof. Clearly, the identity holds for allm,n withγ(m)|γ(n) and (m,n)=1. It thus remains to prove this identity when (m,n)>1. For each (s−1)-excessive pair (m,n) with γ(m)|γ(n), let their prime factorizations be

m=p₁^a1···p^a_u^u, n=p^b₁¹···p^b_u^u, (3.19) wherepiare distinct primes,ainonnegativeintegers, andbipositiveintegers,ai≤bi(i= 1, 2, 3,. . .,u). By multiplicativity, we can write

f(mn)=

u i=1

fp^a_i^i+bi. (3.20)

The right-hand side of the identity becomes

d|(m,n)

g1∗g2

m d

f n

d

µ(d)g1g2

(d)

=

u i=1

ai

j=0

g1∗g2

p^a_i^i−jfp^b_i^i−jµp^jg1g2

p^j.

(3.21)

The identity will be established if we can show that fp^a+b=

a j=0

g1∗g2

p^a−jfp^b−jµp^jg1g2

p^j, (3.22)

for each primepanda∈N∪ {0},b∈Nwithb≥a+s−1. To this end, it suffices to show that

fa+b=g_a^∗fb−g_a^∗₋₁fb−1g₁, (3.23) where f(pⁱ)=fi, (g1∗g2)(pⁱ)=g_i^∗, (g1g2)(p^j)=g_j.

Fora=0, (3.23) trivially holds. Whena=1,b≥s, from Rutkowski’s recurrence, we get

fb+1=c1fb+c2fb−1. (3.24) Noting thatc1=g1^∗,c2= −g1, (3.23) follows in this case. Now proceed by induction on a. Assume that (3.23) holds up toa₋1. Again by Rutkowski’s recurrence, whenb+a_≥ s−1, noting also that f andg1∗g2satisfy the same recurrence, we have

f_a+b=c1f_b+a−1+c2f_b+a−2

=c1

g_a^∗₋₁f_b−g_a^∗₋₂f_b−1g₁+c2

g_a^∗₋₁f_b−1−g_a^∗₋₂f_b−2g₁

=c1g_a^∗₋₁f_b+g_a^∗₋₂f_bc2+c2g_a^∗₋₁f_b−1

=g_a^∗fb−g_a^∗₋₁fb−1g₁,

(3.25)

as required.

Theorem 3.3as stated does not include the caseᏯ(2, 0) because (−1)-excessive pair is not defined. However, going through the above proof, we see that in this case, we simply get the result of Haukkanen referred to in (3.5) above. Since functions inᏯ(2, 0) satisfy the Busche-Ramanujan identity, a natural question to ask is whether aᏯ(3, 0)-function enjoys such property. A trivial example of the identity function I=I∗I∗I=I∗I, which belongs to bothᏯ(2, 0) andᏯ(3, 0), shows that the answer is affirmative in cer- tain cases, whileu∗u∗u=µ₋3∈Ꮿ(3, 0) does not satisfy the Busche-Ramanujan iden- tity. Some necessary conditions forᏯ(3, 0)-functions to satisfy the Busche-Ramanujan identity are given in the next proposition.

Proposition3.4. Let f ∈Ꮿ(3, 0). If f satisfies the Busche-Ramanujan identity f(mn)=

d|(m,n)

f m

d

f n

d

F(d) (m,n∈N), (3.26)

whereF∈ᏹ, then for each primep, there are five possibilities:

(1) f(pⁿ)=0for alln≥1, or (2) f(pⁿ)=(f(p))ⁿfor alln≥1, or

(3) f(p²ⁿ)=(f(p²))ⁿ, f(p²ⁿ⁻¹)=0for alln≥1, or (4) f(pⁿ)=(1 +n)(f(p)/2)ⁿfor alln≥1, or

(5) f(pⁿ)=(1/2)(1 +f(p)/D)((f1+D)/2)ⁿ+ (1/2)(1−f1/D)((f1−D)/2)ⁿfor alln≥ 1, whereD=

4f(p²)−3(f(p))²=0.

Proof. Proceeding as in the proof ofTheorem 3.2, we are looking for necessary conditions for f to satisfy the Busche-Ramanujan identity and this amounts to findingF∈ᏹsuch that

f_a+b= b j=0

f_a−jf_b−jF_j, (3.27)

for each prime p and a≥b, that is, assuming without loss of generality thatνp(m)≥ νp(n). Substitutingb=1 into (3.27), we obtain the main recurrence relation

fa+1= faf1+fa−1F1 (a≥1). (3.28) Puttinga=1, we getF1=f2−f1². FromCorollary 2.7,

f_af1+f_a−1F1=f_af1+ f_a−1

f2−f₁²+f_a−2

f3−2f1f2+ f₁³, (3.29)

which entails

fa−1F1=c2fa−1+c3fa−2 (a≥3), (3.30) wherec2= f2−f1²,c3=f3−2f1f2+f1³. UsingF1=f2−f1²=c2, this last relation simpli- fies toc3f_a−2=0 (a≥3), and so either

(i) fn=0 for alln≥1, or (ii) 0=c3=f3−2f1f2+ f1³.

In the latter situation, we divide into two cases according toc2=0 orc2=0.

Case 1(c2=0). In this case, it easily follows from the main recurrence relation that f_a= f₁^afor alla≥1.

Case 2(c2=0). In this case, we further subdivide into two subcases according to f1=0 or not.

Subcase 2.1(f1=0, and so f2=c2=0). Using the main recurrence relation, it is easily checked that f(p²ⁿ)=(f(p²))ⁿ, and f(p²ⁿ⁻¹)=0 for alln≥1.

Subcase 2.2(f1=0). In this case, the main recurrence relation is a second-order recur- rence with constant coefficients whose characteristic equation isx²−f1x−c2=0, with roots (1/2)(f1±D), whereD=

f1²+ 4c2. The solutions corresponding toD=0 orD=0 are listed as (4) and (5), respectively, in the statement of the proposition.

In the proof ofProposition 3.4,Case 1contains, as a special case, the identity func- tion, while other cases contain some nontrivialᏯ(2, 0)-functions, and some nontrivial Ꮿ-functions.Proposition 3.4indicates somewhat thatᏯ(3, 0)-functions which satisfy rea- sonable Busche-Ramanujan-type identity can be artificially constructed from those sat- isfying conditions in any of the five cases. We now give an example to substantiate this claim. Recall fromCorollary 2.7that f ∈Ꮿ(3, 0)⇔for each primepand integerse≥3, we have

fe+1=fef1+fe−1A+ fe−2B, (2.31) where fe= f(p^e), A=A(p)= f2− f₁²,B=B(p)= f3−2f2f1+ f₁³. Should there be a Busche-Ramanujan-type identity, subject to certain conditions onm,n, proceeding as in the proof ofTheorem 3.2, we deduce that there must existF∈ᏹsatisfying

f_a+b= b i=0

f_a−if_b−iF_i, (2.32)

wherea≥b,Fi=F(pⁱ). Consider theᏯ(3, 0)-function defined by

f(1)=1, f2^a (a≥1) (2.33)

satisfying (2.31) with

B(2)=f2³−2f2²f2¹+f2³=0,

fp^a=0 (a≥1) (2.34)

for all other primesp≥3. This particular function f ∈Ꮿ(3, 0) because it satisfies (2.31) with A(2), A(p), B(p) (p prime ≥3) arbitrary but B(2)=0. It satisfies the Busche- Ramanujan identity (2.32) withF(2)=A(2), F(2ⁱ)=F(p^j)=0 (i≥2, j≥1). The sit- uations for generalᏯ(3,s) andᏯ(r,s) withr≥3 are analogous. The details are omitted.

Another class of identities for functions inᏯ(2, 0), called extended Busche-Ramanujan identity, is due to Redmond and Sivaramakrishnan [16] which states that for f ∈Ꮽ, define

t0(n)=t(n), tk(n)=





t(n) ifn|k,

0 otherwise. (2.35)

LetT0=T,T_k=µ∗t_k. If f =g1∗g2∈Ꮿ(2,0), then

d|(m,n)

f m

d

f n

d g1g2

(d)T_k(d)=

d|(m,n,k)

t(d)g1g2

(d)f mn

d²

. (2.36)

Using exactly the same proof as in [16, Theorem 13], together with the result ofTheorem 3.3, we have the following theorem.

Theorem3.5. Lets∈N. If f =g1∗g2∗h⁻1¹∗ ··· ∗h⁻_s¹∈Ꮿ(2,s), then

d|(m,n)

g1∗g2

m d

f n

d g1g2

(d)T_k(d)=

d|(m,n,k)

t(d)g1g2

(d)f mn

d²

, (3.37)

for each(s−1)-excessive pair(m,n)withγ(m)|γ(n).

4. Binomial-type identities

It is known, see, for example, [16] or [21, Chapter 13], that if f =g1∗g2∈Ꮿ(2, 0), then f satisfies the so-calledbinomial identity

fp^k=

[k/2]

j=0

(−1)^j k−j

j

f(p)^k−2jg1(p)g2(p)^j, (4.1)

wherepis a prime,k∈N.In [6], another form of binomial identity is found, namely,

2^kfp^k=

[k/2]

i=0

k+ 1 2i+ 1

f(p)^k−2if(p)²−4g1(p)g2(p)ⁱ. (4.2)

The derivation of (4.1) in [16] is by induction, while that of (4.2) in [6] is based on solv- ing second-order recurrence relation. Making use of certain Chebyshev-type identities, Haukkanen also derived the following inverse forms of (4.1) and (4.2):

f(p)^k=

[k/2]

i=0

k i

− k

i−1

fp^k−2ig1(p)g2(p)ⁱ, (4.3) (k+ 1)f(p)^k=

[k/2]

i=0

k+ 1 2i

d_2i2^k−2ifp^k−2if(p)²−4g1(p)g2(p)ⁱ, (4.4)

whered2iis defined as in [17, Section 3.4], namely, via the generating series relation 2x

e^x−e^{−x =} ∞ i=0

d2i x²ⁱ

(2i)!. (4.5)