Summation of p-adic Functional Series in Integer Points

Transcription

1 Fiomat 31:5 (2017), DOI /FIL D Pubished by Facuty of Sciences and Mathematics, University of Niš, Serbia Avaiabe at: Summation of p-adic Functiona Series in Integer Points Branko Dragovich a, Andrei Yu. Khrennikov b, Nataša Ž. Mišić c a Institute of Physics, University of Begrade, Begrade, Serbia, and Mathematica Institute of the Serbian Academy of Sciences and Arts, Begrade, Serbia b Internationa Center for Mathematica Modeing in Physics, Engineering, Economics, and Cognitive Science Linnaeus University, Växjö-Kamar, Sweden c Loa Institute, Kneza Višesava 70a, Begrade, Serbia Abstract. Summation of a arge cass of the functiona series, which terms contain factorias, is considered. We first investigated finite partia sums for integer arguments. These sums have the same vaues in rea and a p-adic cases. The corresponding infinite functiona series are divergent in the rea case, but they are convergent and have p-adic invariant sums in p-adic cases. We found poynomias which generate a significant ingredients of these series and make connection between their rea and p-adic properties. In particuar, we found connection of one of our integer sequences with the Be numbers. 1. Introduction The infinite series pay an important roe in mathematics, physics and many other appications. Usuay their numerica ingredients are rationa numbers and therefore the series can be treated in any p-adic as we as in rea number fied, because rationa numbers are endowed by rea and p-adic norms simutaneousy. Hence, for a rea divergent series it may be usefu investigation of its p-adic anaog when p-adic sum is a rationa number for a rationa argument. Many series in string theory, quantum fied theory, cassica and quantum mechanics contain factorias. Such series are usuay divergent in the rea case and convergent in p-adic ones. This was main motivation for considering different p-adic aspects of the series with factorias in [1 11] and many summations performed in rationa points. Aso, using p-adic number fied invariant summation in rationa points, rationa summation [5] and adeic summation [2] were introduced. It is worth mentioning that p-adic numbers and p-adic anaysis have been successfuy appied in modern mathematica physics (from strings to compex systems and the universe as a whoe) and in some reated fieds (in particuar in bioinformation systems, see, e.g. [15]), see [12, 13] for an eary review and [14] for a recent one. Quantum modes with p-adic vaued wave functions, see, e.g. [16] for the recent review, generated various p-adic series eading to nontrivia summation probems (see, e.g. [17 19]) Mathematics Subject Cassification. Primary 11E95, 40A05; Secondary 11D88, 11B83 Keywords. p-adic series, p-adic invariant summation, integer sequences, Be numbers Received: 10 Apri 2015; 16 August 2015 Communicated by Eberhard Makowsky Research supported by Ministry of Education, Science and Technoogica Deveopment of the Repubic of Serbia, projects: OI , TR and TR A part of this work was done during B.D. visit of the Internationa Center for Mathematica Modeing in Physics, Engineering, Economics, and Cognitive Science, Linnaeus University, Växjö, Sweden. Emai addresses: dragovich@ipb.ac.rs (Branko Dragovich), Andrei.Khrennikov@nu.se (Andrei Yu. Khrennikov), nmisic@afrodita.rcub.bg.ac.rs (Nataša Ž. Mišić)

2 B. Dragovich et a. / Fiomat 31:5 (2017), In this paper we consider p-adic invariant summation of a wide cass of finite and infinite functiona series which terms contain factorias, i.e. ε n (n + ν)!p kα (n; x)x αn+β, where ε = ±1, and parameters ν, β N 0 = N {0}, k, α N. P kα (n; x) are poynomias in x of degree kα which coefficients are some poynomias in n. We show that there exist poynomias P kα (n; x) for any degree kα, such that for any x Z vaues of the sums do not depend on p. Moreover, we have found recurrence reations to cacuate such P kα (n; x) and other reevant poynomias. The obtained resuts are generaization of recenty obtained ones for the series n!pk (n; x)x n, see [21]. Some resuts are iustrated by simpe exampes. A necessary genera information on p-adic series can be found in standard books on p-adic anaysis, see, e.g. [20]. 2. Some Functiona Series with Factorias where We consider functiona series of the form S kα (x) = + ε n (n + ν)! P kα (n; x) x αn+β, ε = ±1, ν, β N 0 = N {0}, α, k N, (1) P kα (n; x) = C kα (n) x kα + C (k 1)α (n) x (k 1)α + + C α (n) x α + C 0 (n), i C iα (n) = c ij n jα, 0 i k, c ij Z. (2) j=0 Since rationa numbers beong to rea as we as to p-adic numbers, the series (1) can be considered as rea (x R) as p-adic (x Q p ) ones. In the rea case, (1) is evidenty divergent. In the seque we sha investigate (1) p-adicay Convergence of the p-adic Series Necessary and sufficient condition for the p-adic power series to be convergent [13, 20] coincides, i.e. S(x) = + n=1 a n x n, a n Q Q p, x Q p, a n x n p 0 as n, (3) where p denotes p-adic absoute vaue (aso caed p-adic norm). To prove this assertion, note that p- adic absoute vaue is utrametric (non-archimedean) one and satisfies inequaity x + y p max{ x p, y p }. Now suppose that the series (3) is convergent for some arguments x and the corresponding sum is S(x), i.e S(x) S n (x) p 0 as n, where S n (x) = a 0 + a 1 x a n 1 x n 1. Then a n x n p = S n+1 (x) S n (x) p = S n+1 (x) S(x)+S(x) S n (x) p max{ S(x) S n+1 (x) p, S(x) S n (x) p } 0 as n. That a n x n p 0 as n is sufficient condition foows from the Cauchy criterion. Namey, for enough arge n and arbitrary m, due to utrametricity one can write a n x n p = a n x n + a n+1 x n a n+m x n+m p. The functiona series (1) contains (n + ν)!, hence to investigate its convergence one has to know p-adic norm of (n + ν)!. First, one has to know a power M(n) by which prime p is contained in n! (see, e.g. [13] or [21]). Let n = n 0 + n 1 p n r p r and s n = n 0 + n n r denotes the sum of digits in expansion of a natura number n in base p. Then, one has n! = m p M(n) = m p n sn p 1, p m, n! p = p n sn p 1, (n + ν)! p = p n+ν sn+ν p 1. (4) Theorem 2.1. p-adic series (1) is convergent for every x Z p and any p.

3 B. Dragovich et a. / Fiomat 31:5 (2017), Proof. Consider p-adic norm of the genera term in (1) when x Z p, i.e. ε n (n + ν)! P kα (n; x) x αn+β p (n + ν)! p = p n+ν sn+ν p 1 0 as n, (5) where P kα (n; x) p 1 and x αn+β p 1. Hence, the power series ε n (n + ν)! P kα (n; x) x αn+β is convergent in Z p, i.e. x p 1. Since p Z p = Z, it means that the infinite series ε n (n + ν)! P kα (n; x) x αn+β is simutaneousy convergent for a integers and a p-adic norms. 3. Summation at Integer Points Mainy we are interested for which poynomias P kα (n; x) we have that if x Z then the sum of the series (1) is S kα (x) Z, i.e. S kα (x) is aso an integer which is the same in a p-adic cases. Since poynomias P kα (n; x) are determined by poynomias C iα (n), 0 i k (2), it means that one has to find these C iα (n), 0 i k. Our task is to find connection between poynomia P kα (n; x) and sum of infinite series S kα (x), which becomes aso a poynomia. We are interested now in determination of the poynomias P kα (n; x) and the corresponding sums S kα (x) = Q kα (x) of the infinite series (1), where Q kα (x) = q kα x kα + q (k 1)α x (k 1)α + + q α x α + q 0 (6) are aso some poynomias reated to P (kα) (n; x), so that P (kα) (n; x) and Q (kα) (x) do not depend on concrete p-adic consideration and that they are vaid for a x Z. A very simpe and iustrative exampe [20] of p-adic invariant summation of the infinite series (1) is n! n = 1!1 + 2!2 + 3! = 1 (7) n 0 which obtains taking x = 1, P 11 (n; 1) = n and gives Q 11 (1) = 1. To prove (7), one can use any one of the foowing two properties: n! n = 1 + N!, n!n = (n + 1)! n!. (8) n=1 In the seque we sha deveop and appy approach of partia sums which generaize the first one in (8) The Partia Sums Having in mind our goa on rationa summation of the functiona series (1), et us consider the partia sums of its simpified version. Namey, S k (N; x) = ε n (n + ν)! (n + ν) k x αn+β = ν! ν k x β + ε n (n + ν)! (n + ν) k x αn+β = ν! ν k x β + ε x α ε n (n + ν)! (n + ν + 1) k+1 x αn+β ε N (N + ν)! (N + ν) k x αn+β n=1 = ν! ν k x β + ε x α ε n (n + ν)! (n + ν) x αn+β ε N (N + ν)! (N + ν) k x αn+β =0 = ν! ν k x β + ε x α S 0 (N; x) + ε x α S (N; x) ε N (N + ν)! (N + ν) k x αn+β, (9) =1

4 B. Dragovich et a. / Fiomat 31:5 (2017), where S 0 (N; x) = N 1 ε n (n + ν)! x αn+β. Obtained recurrence reation (9) gives possibiity to find sums S k (N; x), k N, with respect to S 0 (N; x). Performing operations for k = 0 and k = 1 in (9), one obtains S 1 (N; x) = (ε x α 1) S 0 (N; x) ε ν! x β α + ε n 1 (N + ν)! x αn+β α, (10) S 2 (N; x) = ((ε x α 2)(ε x α 1) 1) S 0 (N; x) + ε ν! x β α (2 ε x α ν) + (ε x α 2 + N + ν) ε n 1 (N + ν)! x αn+β α. (11) Equations (10) and (11) can be rewritten in equivaent and more suitabe form, respectivey: ε n (n + ν)![x α (n + ν) + x α ε] x αn+β = ε ν! x β + ε N 1 (N + ν)! x αn+β, (12) ε n (n + ν)![x 2α (n + ν) 2 (x 2α 3 ε x α + 1)] x αn+β = ε ν! [(2 ν)x α ε] x β + [(N + ν 2) x α + ε] ε N 1 (N + ν)! x αn+β. (13) Theorem 3.1. The recurrence reation (9) has soution in the form ε n (n + ν)! [(n + ν) k x kα + U kα (x)] x αn+β = V (k 1)α (x) + A (k 1)α (N; x) ε N 1 (N + ν)! x αn+β, (14) where poynomias U kα (x), V (k 1)α (x) and A (k 1)α (N; x) satisfy the foowing recurrence reations: x (k +1)α U α (x) ε U kα (x) x (k+1)α = 0, U 1α (x) = x α ε, k = 1, 2,..., (15) =1 x (k +1)α V ( 1)α (x) ε V (k 1)α (x) + ε ν! ν k x kα+β = 0, V 0 (x) = ε ν! x β, k = 1, 2,..., (16) =1 x (k +1)α A ( 1)α (N; x) ε A (k 1)α (N; x) (N + ν) k x kα = 0, A 0 (N; x) = 1, k = 1, 2,.... (17) =1 Proof. Formua (14) can be rewritten as S k (N; x) = x kα U kα (x)s 0 (N; x) + x kα V (k 1)α (x) + A (k 1)α (N; x)x kα ε N 1 (N + ν)!x αn+β. (18) Now one can repace S k (N; x) in recurrence reation (9) by this one in (18). Compiing the terms separatey with S 0 (n; s), then with x αn+β and finay a the rest terms, we obtain respectivey recurrence reations for U kα (x), A (k 1)α (N; x) and V (k 1)α (x). Note that factor x β does not pay an important roe in (14), because V (k 1)α (x) aso contains x β and it can be excuded from this formua by redefinition of V (k 1)α (x). Theorem 3.2. Poynomias U kα (x) and V (k 1)α (x) are reated to poynomia A (k 1)α (N; x) in the form U kα (x) = (ν + 1)x α A (k 1)α (1; x) εa (k 1)α (0; x) ν k x kα, k N, (19) V (k 1)α (x) = ε ν! x β A (k 1)α (0; x), k N. (20) Proof. We use equation (14). Note that U kα (x) and V (k 1)α (x) do not depend on the upper imit of summation in (14). Hence, subtracting equations in (14) with N 1 and N 2, we obtain reation (N + ν 1) k x kα + U kα (x) = (N + ν) x α A (k 1)α (N; x) ε A (k 1)α (N 1; x) (21)

5 B. Dragovich et a. / Fiomat 31:5 (2017), which does not contain V (k 1)α (x). Taking N = 1 in (21), one obtains expression (19) for U kα (x). Now using (14) when N = 1 gives ν! ν k x kα+β + U kα (x) x β ν! = V (k 1)α (x) + A (k 1)α (1; x) (ν + 1)! x α+β. (22) Combining (21) and (22), it foows (20). Recurrent formuas (15) (17) enabe to cacuate poynomias U kα (x), V (k 1)α (x) and A (k 1)α (N; x) for any k N, knowing initia expressions: U 1 (x) = x α ε, V 0 (x) = ε ν! x β and A 0 (N; x) = 1. For the first five vaues of degree k, we have obtained the foowing expicit expressions. k = 1 U 1α (x) =x α ε, V 0 (x) = ε ν! x β, A 0 (n; x) =1. (23) k = 2 U 2α (x) = x 2α + 3εx α 1, V 1α (x) = εν!x β [(ν 2)x α + ε], A 1α (n; x) =(n + ν 2)x α + ε. (24) k = 3 U 3α (x) =x 3α 7εx 2α + 6x α ε, V 2α (x) = εν!x β [(ν 2 3ν + 3)x 2α + (ν 5)εx α + 1], A 2α (n; x) =[(n + ν) 2 3(n + ν) + 3]x 2α + (n + ν 5)εx α + 1. (25) k = 4 U 4α (x) = x 4α + [ν 3 (1 ε) 4ν 2 (1 ε) + 6ν(1 ε) ε]x 3α + [ν 2 (1 ε) 7ν(1 ε) 8 17ε]x 2α + 10εx α 1, V 3α (x) = εν!x β [(ν 3 4ν 2 + 6ν 4)x 3α + (ν 2 7ν + 17)εx 2α + (ν 9)x α + ε], A 3α (n; x) =[(n + ν) 3 4(n + ν) 2 + 6(n + ν) 4]x 3α + [(n + ν) 2 7(n + ν) + 17]εx 2α + (n + ν 9)x α + ε. (26) k = 5 U 5α (x) =x 5α (ν )εx 4α + 90x 3α 65εx 2α + 15x α ε, V 4α (x) = εν!x β [(ν 4 5ν ν 2 10ν + 5)x 4α + (ν 3 9ν ν 49)εx 3α + (ν 2 12ν + 52)x 2α + (ν 14)εx α + 1], A 4α (n; x) =[(n + ν) 4 5(n + ν) (n + ν) 2 10(n + ν) + 5]x 4α + [(n + ν) 3 9(n + ν) (n + ν) 49]εx 3α + [(n + ν) 2 12(n + ν) + 52]x 2α + (n + ν 14)εx α + 1. (27)

6 B. Dragovich et a. / Fiomat 31:5 (2017), It is worth emphasizing that a the above equaities, in particuar (9) and (14), are vaid in rea and a p-adic cases. The centra roe in (14) pays poynomia A kα (N; x), which is soution of the recurrence reation (17), because poynomias U kα (x) and V (k 1) (x) are simpy connected to A kα (N; x) by formuas (19) and (20), respectivey. When N in (14), the term with poynomia A (k 1)α (N; x) p-adicay vanishes giving the sum of the foowing p-adic infinite functiona series: ε n (n + ν)! [(n + ν) k x kα + U kα (x)] x αn+β = V (k 1)α (x). (28) This equaity has the same form for any k N, and poynomias U kα (x) and V (k 1)α (x) separatey have the same vaues in a p-adic cases for any x Z. In other words, nothing depends on particuar p-adic properties in (28) when x Z, i.e. this is p-adic invariant resut. This resut gives us the possibiity to present a genera soution of the probem posed on p-adic invariant summation of the series (1). Theorem 3.3. The functiona series (1) has p-adic invariant sum if S kα (x) + ε n (n + ν)! P kα (n; x) x αn+β = Q kα (x) (29) P kα (n; x) = where B j, x Z. B j [(n + ν) j x jα + U jα (x)] and Q kα (x) = j=1 B j U jα (x), (30) j=1 Note that A kα (n; x) as we as U kα (x) and V (k 1)α (x) can be written in the compact form A kα (n; x) = A (kα) (n + ν) x α, U kα (x) = =0 U (kα) x α, V kα (x) = =0 V (kα) x α, (31) =0 where A (kα) (n + ν) is a poynomia in n + ν of degree with (n + ν) as the term of the highest degree. Putting x = 0 in (15) (17), the foowing properties hod: A kα (n; 0) = εa (k 1)α (n; 0) = ε k, k = 1, 2,... U (k+1)α (0) = εu kα (0) = ε k+1, k = 1, 2,... V kα (0) = εv (k 1)α (0) = ν!x β ε k+1, k = 1, 2,... As an iustration of summation formua (28), we present five simpe (k = 1,...,5) exampes. k = 1 k = 2 ε n (n + ν)! [(n + ν + 1)x α ε] x αn = ε ν!, x Z. (32) ε n (n + ν)! {[(n + ν) 2 1]x 2α + 3εx α 1} x αn = ε ν! [(2 ν)x α ε], x Z. (33)

7 B. Dragovich et a. / Fiomat 31:5 (2017), k = 3 k = 4 k = 5 ε n (n + ν)! {[(n + ν) 3 + 1]x 3α 7εx 2α + 6x α ε} x αn = ε ν![(ν 2 3ν + 3)x 2α + (ν 5)εx α + 1], x Z. (34) ε n (n + ν)! { [(n + ν) 4 1]x 4α + [ν 3 (1 ε) 4ν 2 (1 ε) + 6ν(1 ε) ε]x 3α + [ν 2 (1 ε) 7ν(1 ε) 8 17ε]x 2α + 10εx α 1 } x αn = ε ν! [(ν 3 4ν 2 + 6ν 4)x 3α + (ν 2 7ν + 17)εx 2α + (ν 9)x α + ε], x Z. (35) ε n (n + ν)! { [(n + ν) 5 + 1]x 5α (ν )εx 4α + 90x 3α 65εx 2α + 15x α ε } x αn = ε ν! [(ν 4 5ν ν 2 10ν + 5)x 4α + (ν 3 9ν ν 49)εx 3α + (ν 2 12ν + 52)x 2α + (ν 14)εx α + 1], x Z. (36) 4. Discussion and Concuding Remarks The main resuts presented in this paper are summation formua (9) and theorems (3.1) (3.3). These resuts are generaizations of some earier resuts, see [9 11, 21]. Finite series (9) with their sums (14) are vaid for rea and p-adic numbers. When n the corresponding infinite series are divergent in rea case, but are convergent and have the same sums in a p-adic cases. This fact can be used to extend these sums to the rea case. Namey, the sum of a divergent series depends on the way of its summation and here it can be used its integer sum vaid in a p-adic number fieds. This way of summation of rea divergent series was introduced for the first time in [2] and caed adeic summabiity. An importance of this adeic summabiity depends on its potentia future use in some concrete exampes. The simpest infinite series with n! is n!. It is convergent in a Z p, but has not p-adic invariant sum. Even it is not known so far does it has a rationa sum in any Z p. Rationaity of this series and n!n k x n was discussed in [9]. The series n! is aso reated to Kurepa hypothesis which states (!n, n!) = 2, 2 n N, where!n = n 1 j=0 j!. Vaidity of this hypothesis is sti an open probem in number theory. There are many equivaent statements to the Kurepa hypothesis, see [10] and references therein. From p-adic point of view, the Kurepa hypothesis reads: j=0 j! = n 0 + n 1 p + n 2 p 2 +, where digit n 0 0 for a primes p 2. It is worth emphasizing that poynomias A kα (n; x) contain a information about properties of series (14). For various combinations of x = 0, ±1, ±2,..., n = 0, 1, 2,... and parameters k, α, N one can obtain integer sequences, and some of them are aready known. Note that parameter ν in poynomias A kα (n; x) appears in the form n + ν and it is enough to consider how A kα (n; x) depends on n. Hence we wi consider A kα (n; x) with parameter ν = 0. Here are some simpe sequences derived from A kα (n; x). α = any even natura number : A kα (0; ±1) : 1, ε 2, 4 5ε, ε, 58 63ε,... k N 0, ε = ±1, A kα (1; ±1) : 1, 1 + ε, 2 4ε, ε, 43 39ε,... k N 0, ε = ±1. α = any odd natura number : A kα (0; 1) : 1, ε 2, 4 5ε, ε, 58 63ε,... k N 0, ε = ±1, A kα (1; 1) : 1, 1 + ε, 2 4ε, ε, 43 39ε,... k N 0, ε = ±1.

8 α = any odd natura number : B. Dragovich et a. / Fiomat 31:5 (2017), A kα (0; 1) : 1, 2 + ε, 4 + 5ε, ε, ε,... k N 0, ε = ±1, A kα (1; 1) : 1, 1 + ε, 4ε, ε, ε,... k N 0, ε = ±1. Beow are aso some simpe integer sequences derived from V kα (x) and U kα (x). x = 1, α N, β = 0, ν = 0 : V kα (1) : ε, 1 + 2ε, 5 4ε, ε, 63 58ε,... k N 0, ε = ±1, U kα (1) : 1 ε, 2 + 3ε, 7 8ε, 1 3ε, ε,... k N, ε = ±1. x = 1, α N, β = 0, ν = 1 : V kα (1) : ε, 1 + 2ε, 5 4ε, ε, 63 58ε,... k N 0, ε = ±1, U kα (1) : 1 ε, 2 + 3ε, 7 8ε, ε, 2,... k N, ε = ±1. When x = ±1, ε = α = 1, β = ν = 0, then (28) becomes n! [n k + u k ] = v k if x = 1, ( 1) n n! [( 1) k+1 n k + ū k ] = v k if x = 1, (37) where u k = U k1 (1), v k = V (k 1)1 (1) and ū k = U k1 ( 1), v k = V (k 1)1 ( 1) are some integers. First equaity in (37) was introduced in [5], and properties of u k and v k are investigated in series of papers by Dragovich (see references [8 11]). In [22] some reationships of u k with the Stiring numbers of the second kind are estabished, and p-adic irrationaity of n 0 n!n k was discussed (see [23 25]). Note that the foowing sequences are reated to some rea (combinatoria) cases, compare with [26]: v k = A (k 1)1 (0; 1) = V (k 1)1 (1) : 1, 1, 1, 5, 5, 21, 105, 141,... see A (38) u k = A (k 1)1 (1; 1) A (k 1)1 (0; 1) = U k1 (1) : 0, 1, 1, 2, 9, 9, 50, 267,... see A (39) ū k = A (k 1)1 (1; 1) + A (k 1)1 (0; 1) = U (k1 ( 1) : 2, 5, 15, 52, 203, 877, 4140, 21147,... see A (40) v k = A (k 1)1 (0; 1) = V (k 1)1 ( 1) : 1, 3, 9, 31, 121, 523, 2469, 12611,... see A (41) It is worth pointing out integer series (40) and (41), which are directy cacuated from the foowing recurrence reations: ū k+1 = ( 1) k ū + ū k + ( 1) k, ū 1 = 2, k = 1, 2, 3,..., (42) =1 v k+1 = ( 1) k v + v k, v 1 = 1, k = 1, 2, 3,.... (43) =1 In particuar, the series of integers ū k, (k = 1, 2, 3,...) coincides with the Be numbers B k, (k = 0, 1, 2,...) by equaity B k+1 = ū k for k 1 (at east for the first 8 terms directy cacuated). Reca that the Be numbers B k are equa to the number of partitions of a set of k eements. They satisfy the recurrence reation B k+1 = ( ) k B, B 0 = 1. =0 It foows that the number of partitions of sets with more than one eement can be obtained aso from the recurrence reation for ū k given by (42). Various aspects of the poynomias A kα (n; x), (k = 0, 1, 2,..., α = 1, 2, 3,...) deserve to be further anayzed.

9 B. Dragovich et a. / Fiomat 31:5 (2017), References [1] I. Ya. Arefeva, B. Dragovich and I. V. Voovich, On the p-adic summabiity of the anharmonic osciator, Phys. Lett. B 200, (1988). [2] B. Dragovich, p-adic perturbation series and adeic summabiity, Phys. Lett. B 256 (3,4), (1991). [3] B. G. Dragovich, Power series everywhere convergent on R and Q p, J. Math. Phys. 34 (3), (1992) [arxiv:mathph/ ]. [4] B. G. Dragovich, On p-adic aspects of some perturbation series, Theor. Math. Phys. 93 (2), (1993). [5] B. G. Dragovich, Rationa summation of p-adic series, Theor. Math. Phys. 100 (3), (1994). [6] B. Dragovich, On p-adic series in mathematica physics, Proc. Stekov Inst. Math. 203, (1994). [7] B. Dragovich, On p-adic series with rationa sums, Scientific Review 19 20, (1996). [8] B. Dragovich, On some p-adic series with factorias, in p-adic Functiona Anaysis, Lect. Notes Pure App. Math. 192, (Marce Dekker, 1997) [arxiv:math-ph/ ]. [9] B. Dragovich, On p-adic power series, in p-adic Functiona Anaysis, Lect. Notes Pure App. Math. 207, (Marce Dekker, 1999) [arxiv:math-ph/ ]. [10] B. Dragovich, On some finite sums with factorias, Facta Universitatis: Ser. Math. Inform. 14, 1 10 (1999) [arxiv:math/ [math.nt]]. [11] M. de Gosson, B. Dragovich and A. Khrennikov, Some p-adic differentia equations, in p-adic Functiona Anaysis, Lect. Notes Pure App. Math. 222, (Marce Dekker, 2001) [arxiv:math-ph/ ]. [12] L. Brekke and P. G. O. Freund, p-adic numbers in physics, Phys. Rep. 233, 1 66 (1993). [13] V. S. Vadimirov, I. V. Voovich and E. I. Zeenov, p-adic Anaysis and Mathematica Physics (Word Sci. Pub., Singapore, 1994). [14] B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Voovich, On p-adic mathematica physics, p-adic Numbers Utram. Ana. App. 1 (1), 1 17 (2009) [arxiv: [math-ph]]. [15] B. Dragovich and A. Yu. Dragovich, A p-adic mode of DNA sequence and genetic code, p-adic Numbers Utram. Ana. App. 1 (1), (2009) [arxiv:q-bio/ [q-bio.gn]]. [16] S. Abeverio, R. Cianci and A. Yu. Khrennikov, p-adic vaued quantization, p-adic Numbers, Utrametric Anaysis and Appications 1 (2), (2009). [17] S. Abeverio, A. Khrennikov and R. Cianci, On the spectrum of the p-adic position operator, J. Physics A: Math. and Genera 30, (1997). [18] S. Abeverio, A. Khrennikov and R. Cianci, On the Fourier transform and the spectra properties of the p-adic momentum and Schrodinger operators, J. Physics A: Math. and Genera 30, (1997). [19] S. Abeverio, A. Khrennikov and R. Cianci, A representation of quantum fied hamitonian in a p-adic Hibert space, Theor. Math. Physics 112 (3), (1997). [20] W. H. Schikhof, Utrametric Cacuus: An Introduction to p-adic Anaysis (Cambridge Univ. Press, Cambridge, 1984). [21] B. Dragovich and N. Z. Misic, p-adic invariant summation of some p-adic functiona series, p-adic Numbers Utr. Ana. App. 6 (4), (2014), [arxiv: v1 [math.nt]]. [22] M. Ram Murty and S. Sumner, On the p-adic series n=1 nk n!, in Number Theory, CRM Proc. Lecture Notes 36, (Amer. Math. Soc., 2004). [23] P. K. Saikia and D. Subedi, Be numbers, determinants and series, Proc. Indian Acad. Sci. (Math. Sci.) 123 (2), (2013). [24] D. Subedi, Compementary Be numbers and p-adic series, J. Integer Seq. 17, 1 14 (2014). [25] N. C. Aexander, Non-vanishing of Uppuuri-Carpenter numbers, Preprint [26] N. J. A. Soane, The on-ine encycopedia of integer sequences,