The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked

Applied Mathematics, 2019, 10, 100-112 http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385

Do Tan Si

The HoChiMinh-City Physical Association, HoChiMinh-City, Vietnam

How to cite this paper: Si, D.T. (2019) The Abstract Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked. Applied Mathemat- Utilizing translation operators we get the powers sums on arithmetic pro- ics, 10, 100-112. gressions and the Bernoulli polynomials of order m under the form of diffe- https://doi.org/10.4236/am.2019.103009 rential operators acting on monomials. It follows that (ddnz− dd) applied

Received: February 18, 2019 on a power sum has a meaning and is exactly equal to the Bernoulli poly- Accepted: March 19, 2019 nomial of the same order. From this new property we get the formula giving Published: March 22, 2019 powers sums in term of sums of successive derivatives of Bernoulli polynomi-

Copyright © 2019 by author(s) and al multiplied with primitives of the same order of n. Then by changing the Scientific Research Publishing Inc. two arguments zn, into Z= zz( −1) , λ where λ designed the 1st order This work is licensed under the Creative Commons Attribution International power sums and proving that Bernoulli polynomials of odd order vanish for License (CC BY 4.0). arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for pow- http://creativecommons.org/licenses/by/4.0/ ers sums in term of polynomials in λ having coefficients depending on Z. Open Access These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Ber- noulli numbers, the Bernoulli polynomials, the powers sums and the Faulha- ber formula for powers sums.

Keywords Bernoulli Numbers, Bernoulli Polynomials, Powers Sums, Faulhaber Conjecture, Shift Operator, Operator Calculus

1. Introduction

The problem of calculating the sums of the mth powers of n first integers n ∑∑nkmm= (1.1) k =1

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is investigated from antiquity by mathematicians around the world. We learn for examples in the thesis of Coen [1] as so as in Edwards [2] that at the beginning of the 11th century ibn al-Haytham had developed the formulae nn2 nnn32 nnn432 ∑ n = + , ∑ n2 =++, ∑ n3 =+− (1.2) 22 3 26 424 In 15th century his successors had found nnn543 n ∑ n4 =++− (1.3) 5 2 3 30 About two centuries quietly passed until the day in 1631 when Faulhaber [3] published at Ulm the prodigious results for sums of odd powers from ∑ n until ∑ n17 in term of powers of ∑ n . One may find more details on the works of Faulhaber in the reference [4]. After Faulhaber, in 1636 French mathematicians Fermat utilizing the figurate numbers and in 1656 Pascal utilizing results of arithmetic triangle, found also recurrence formulae for calculating ∑ n m from lower-order sums [5]. Then in 1713 in his posthumous Ars conjectandi, Jacob Bernoulli [6], men- tioning Faulhaber, published the lists of ten first ∑ n m . It is plausible that from m this list he observed that ∑ n may be written in terms of the numbers Bk which are the same for all m m m 1 j m +1 mj+−1 ∑∑n =( −) Bnj . (1.4) m +1 j=0 j This famous conjectured formula of Bernoulli was proven in 1755 based on the calculus of finite difference by Euler [7], a researcher working with the Ber- noulli brothers at Zurich. As informed by Raugh [8], Euler had followed de

Moivre in given Bk the denomination Bernoulli numbers, introduced the Ber- noulli polynomials in 1738 via the generating function t ∞ 1 zt = m t e ∑ Bm ( zt) (1.5) e1− m=0 m!

Returning to the Faulhaber conjecture saying that Snm ( ) is a polynomial in

Sn1 ( ) for all m m (m) j Snmj( ) = ∑ ASn1 ( ) (1.6) j=1

we know that Jacobi [9] has the merit of giving the right proof for this conjecture (m) and moreover calculating the first six Faulhaber coefficients Aj although he did not get a formula for obtaining all of them. Another merit of Jacobi consists in pioneering the use of the derivative with respect to n when observing that

m 1 ∑ n=( Bnmm++11( +−10) B( )) (1.7) m +1

d mm−1 ∑∑n= Bnmm( ) = mn + B (1.8) dn Long years passed until Edwards [10] showed how to obtain the Faulhaber

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coefficients by matrix inversion and Knuth [4] by identification of coefficients of odd powers of n in Bernoulli formula with those in Faulhaber conjecture. Nev- ertheless the methods of Edwards and Knuth are not easy to apply. Following Coen, we know the existence of the work Bernoulli numbers: bibliography (1713-1990) of Dilcher [11] which contained 1956 references by 839 authors! Concerning the more general problem of powers sums on arithmetic progressions

m m m Sm ( zn,1) = z ++( z) +++− ( z( n 1)) (1.9)

we remark the recent formula given by Dattoli, Cesarano, Lorenzutta [12] in term of Bernoulli polynomials 1 Szn( , ) =( BznBz++( +−) ( )) (1.10) mm +1 mm11

and the formula of Chen, Fu, Zhang [13] in term of sums of powers of S1 ( zn, )

m j 2m 1 2 2mS21m−−( zn ,) =∑ B22mj(2, S 1( zn) z+− B1( z)) B2m( z) (1.11) j=0 2 j 2

which are not easy to apply. After these authors we have proposed a method leading to the formula [14]

k +1 1 ˆ( j) j S21kk++( zn,) = ∑ S21( Z)(2, S 1( zn)) (1.12) j=1 j! ˆ where Z= zz( −1) and SZmm( ) ≡ Sz( ) . The Faulhaber formula is thus obtained but we see that the method for ob- ˆ taining it is cumbersome and the practice calculations of SZmm( ) ≡ Sz( ) for obtaining the Faulhaber coefficients fastidious. Rethinking the problem, we observe that an arithmetic progression is a matter

of translation, that there is a somehow symmetry between nz, and SnZ1 ( ),

and ∂∂nz, so that finally we found a more concise method for resolving the problem and theoretically and practically that we will expose in the following paragraphs.

2. Representation of a Power Sums and Bernoulli Polynomials by Differential Operators

Let

m m m Sm ( zn,1) = z ++( z) +++− ( z( n 1)) n−1 m (2.1) =+∈∑( zk) , zC k =0

be the powers sums on an arithmetic progression and

mm m Snmm( ) ≡ S(0, n) = 0 +++− 1 (n 1) (2.2)

the powers sums of the first integers.

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We may utilize the shift or translation operator ea∂z fz( ) = fz( + a) (2.3)

to get the differential representation

n∂z ∂ (n−∂1) e1− = +z + +z mm = ∀∈ Sm ( zn,) ( 1e e ) z∂ z, n N (2.4) e1z − which gives directly the relations

n∂z e1− m ∂=∂zmS( zn, ) z z (2.5) e1∂z −

∂=zmS( zn,,) mSm−1 ( zn) (2.6)

and the generating function ∞∞tm en∂z −− 1ztm m e1 nt S( zn,e) = = zt (2.7) ∑∑m ∂z t mm=00mm!!e1− = e1−

Because (2.4) is valid for all integers n it is also valid for all real and complex values so that we may write

n∂z ∂ z e m ∂=nmS( zn, ) z (2.8) e1∂z −

Defining now the set of polynomials Bzm ( ) by the differential representation

∂ z m Bzm ( ) = z (2.9) e1∂z −

we see that Bzm ( ) verify m Bzm( +−1) Bz mz( ) =∂ z (2.10)

Bmm′ ( z) = mB−1 ( z) (2.11)

and have the generating function ∞ ttm 1 t =zt =zt = −+ + + ∑ Bm ( z) t e e1( 1zt ) (2.12) m=0 m!2e1− t  1++ 2 allowing the identification of them with Bernoulli polynomials defined by Euler [7] and giving the first of them 1 Bz( ) = 1, Bz( ) = z − . (2.13) 0 1 2 The Bernoulli polynomials are linked to powers sums according to formulae (2.5), (2.8), (2.9) by the relations ∂ nn∂∂zzz m ∂zmSzn( ,) =( e1 −) ∂ z =( e1 −) Bzm( ) = Bzn m( +−) Bzm( ) (2.14) e1z −

z∂n ∂=nmS( zn,e) Bm( n) =+ B m( z n) (2.15)

which lead to the followed beautiful formula where the second member does not depend on n

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(∂n −∂ zm) S( zn, ) = B m( z) (2.16)

Besides it leads also to the formula

∂=nmS( zn,,) mSm−1 ( zn) + Bm( z) (2.17)

which jointed with (2.15) gives rise to the historic Jacobi conjectured formula [9]

∂nmS( n) = mSm−1 ( n) += Bm(0) B m( n) (2.18)

From (2.18) we see that Bm (0) are identifiable with Bernoulli numbers Bm .

3. The Powers Sums

1) Powers sums in terms of Bernoulli polynomials and powers of n

From the Equation (2.16) and the boundary condition Szm ( ,0) = 0 we get immediately the solution of (2.16) 1 S( zn, ) = B( z) mm∂ −∂ nz (3.1) m kk− =∑ ∂∂znB m( zn) k =0

which may be put under the algorithmic form very easy to remember

21m+ (1) nn(m) S( zn, ) = B( z) n + B( z) ++ B( z) (3.2) m mm2! m(m +1) !

or taking into account (2.11) nk +1 S( zn,) = B( z) n + + mm( −11) ( m −+ k) B− ( z) m m mk (k +1!)

nm+1 ++ mB! ( z) 0 (m +1!)

Putting z = 0 in (3.2) and replacing Bmk− (0) with Bmk− we recognize the famous Bernoulli formula [6] m nnkm++11 Sm( n) = Bn m ++ Bmk− ++B0 (3.3) k km++11

2) The Faulhaber formula on powers sums

In Sm ( zn, ) instead of utilizing z and n for arguments let us utilize 1 n2 Z= zz( −1) and λ =S1 ( zn, ) =−+ z n (3.4) 22

Because dZZ ddλλ d =2Bz11( ) , ==+= 0, Bz( ) n, n (3.5) dz dd nn d z

we have

∂≡n (Bz1 ( ) + n) ∂λ (3.6)

∂≡zZ2Bz1 ( ) ∂+∂ nλ (3.7)

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∂−∂≡nzBz11( ) ∂−λ 2 Bz( ) ∂Z (3.8) ≡Bz1 ( )( ∂−∂λ 2 Z )

Equation (2.16) becomes

−1 (∂−∂λ 2,Zm) S( zn) = B1 ( z) Bm( z) (3.9)

Happily from the definition of Bzm ( ) by differential operators (2.9) we may write down

−∂ z m Bzm (−=) ( − z) e1−∂z −

∂ −∂ m =e z z ( −z) (3.10) 1e− ∂z m =−+( ) Bzm ( 1)

which jointed with (2.10) leads to the important properties 1 B21k+=0, B21 k +( 1) =−=+ BB 21 k ++( 0) 21 kk( 0) δ ,0 (3.11) 2

20B1 ( ) = − 1 (3.12)

−1 B1( zB) 21k + ( z) =0 for z = 0, z = 1, k > 0 (3.13)

As an polynomial of order 2k having zz01, for roots may be put by identification of coefficients under the form of a homogeneous polynomial of order k in

( zz−−01)( zz) we obtain the important property: −1 B1( zB) 21k + ( z) is a homogeneous polynomial of order k in Z= zz( −1) k > 0 (3.14)

By the way we notify that because Sn21k + ( ) = 0 for n = 0,1 it is also a homogeneous polynomial of order (k +1) in u= nn( −1) as conjectured Faulha- ber and proven somehow by Jacobi. Moreover the calculations of the quoted polynomials in Z or in u may be done thank to the hereinafter Table 3. Jointed (3.14) with the formula came from the Jacobi formula (2.18) −1 2∂=ZkS21++( z) B 1( zB) 21k( z) (3.15) ˆ we may define a polynomial SZk ( ) of order k depending on Z such that ˆ SZSkk++1( ) ≡ 21( z) (3.16) ˆ −1 2∂≡ZkS++1( Z) B 1( zB) 21k ( z) (3.17) ˆ The definition of SZk ( ) by (3.16) differs a little in index with its definition in [15]. Thank to these considerations we get the solution of (3.9) corresponding to

the boundary condition Szm ( ,0) = 0 under the form −1 ˆ S21k++( zn,) =∂−∂( λ 22Zk) SZ′ 1( ) k (3.18) jj+−1 ˆ =∂∂∑ ZkSZ+12( ) λ (2λ) j=0

or under the algorithmic form

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λλ12 λk +1 ˆ (22) ˆ ( ) ˆ(k +1) ( 2) S++( znSZ, ) =′ ( ) + SZ′′ +( ) ++ S+( Z) (3.19) 21kk1 1! k1 2! k1 (k +1) !

where we see that the Faulhaber coefficients are successive derivations beginning from the first one of the power sums on integers writing under the form ˆ SZSkk++1( ) ≡ 21( z) . Curiously by replacing z with n and consequently Z with u= nn( −1) in the ˆ definition SZSkk++1( ) ≡ 21( z) we get the very important and very interesting formula linking the Faulhaber powers sums on integers and on arithmetic progressions ˆ S21kk++( nSu) = 1( ) (3.20)

For examples ZZ3 2 uu32 nn 655 nn 42 SZˆ ( ) = − ⇒Sn( ) =−=−+ − (3.21) 356 12 6 12 6 2 12 12 uuu432 Z4 ZZ 3 2 Sn( ) =−+⇒SZˆ ( ) = − + (3.22) 748 6 12 8 6 12

4. Obtaining Practically Bernoulli Numbers, Bernoulli Functions and Powers Sums 4.1. Calculations of Bernoulli Numbers

For calculating Bm we remark that from

z∂ 1 (k ) Bzy( +) =e y By( ) = By( ) ++ zBykm( ) ++ zBy( ) m mm k! m 0

m km Bm( z) = B m ++ zBmk− ++ zB0 (4.1) k and (2.10) we get the recursion relation mm  BBm(10) − m( ) =δ m11 = Bm−− ++  Bmk ++ B0 (4.2) 1 k which may be written under the matrix form 1  B 1 12  0 B 0 1 33  1 B 0 1 4 64 2 = (4.3) B 0    3   mm++11 m+1  Bm 0 01 m This matrix equation may be resolved by doing by hand or by program linear combinations over lines from the second one in order to replace them with lines each containing only some non-zero rational numbers.

For instance for calculating successively {BBBBB0,,,,,, 1 2 4 6 B 18} we may utilize the matrix equation

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Matrix equation for calculating Bm . 1 B0 1 12 B1 0 01 3 B2 0

0 0 1 5 B4 0

0 0− 1 0 7 B6 0

0 0 9 5 0 0 9 B8 = 0 (4.4)

0 0− 5 0 0 0 11 B10 0

−61 B12 0 0 3 0 0 0 0 13 105

0 35 0 0 0 0 0 0 15 B14 0

0 240 0 17 0 0 0 0 0 17 B16 0 0 0 2052 0 0− 775 0 0 0 0 0 19 B18

We remark that the last line of this matrix has replaced the line 19 ,i = 0,1,2, 4,6,8,10,12,14,16,18 Some results are i

B0 = 1 , BB01+=20, BB12+=30, −+BB2670 = 1026 775 43867 2052B− 775 BB + 19 ==−− 0 +=−+BB 1 6 18 19 19× 4218 798 18

4.2. Calculations of Bernoulli Polynomials

For calculating Bernoulli polynomials, we remark that (2.11) and the Jacobi formula (2.18) give rise to the relations

z Bz( ) = mB( ttB)d + (4.5) m∫0 mm−1

n Sn( ) = Btt( )d (4.6) mm∫0

which, knowing Bz0 ( ) = 1, permits the easy calculations of all the Bernoulli po-

lynomials and the powers sums on integers Snm ( ) from the set of Bernoulli

numbers {BB01,,, Bm } as we may see hereafter in Table 1.

4.3. Calculations of Powers Sums as Polynomials in n

For calculating Sm ( zn, ) we utilize the algorithmic formula (3.2) and get the results shown in Table 2.

4.4. Calculations of Faulhaber Powers Sums ˆ Practically for transforming Sz21k+ ( ) into SZk+1 ( ) from which one obtains the Faulhaber coefficients let us remark that

22kk++ 1 kk++1121k+ 2k k  k + 1k+1 zZ= + z −  z + +−( ) z (4.7) 12  k + 1

22k+ k+1 so that we may replace the first term z of Sz21k+ ( ) with Z minus a polynomial in z. The polynomial in z so obtained must begin with a term in z 2k and not z 21k+ so that we may continue to replace in it z 2k with a term in Z k

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Table 1. Obtaining Bzm ( ) , Snm ( ) from Bm .

zn B Bz()= mB− () z+= B Sn()() Bn m m∫∫00 mm1 m m n B= 1 Bz ()= 1 Sn()= 00 01 11 nn2 B=−=− Bz() z Sn()=− 112 2 122 11 nnn32 B= Bz() = z2 −+ z Sn()= − + 226 6 23 26 2 432 3 3zz nnn B33=0 Bz () =− z + Sn3()=−+ 22 424 −11nnn543 n B= Bz() =− z4 2 z 32 +− z Sn()= − + − 4430 30 45 2 3 30 55z nn655 nn 42 B=0 Bz () =− z543 z + z − Sn()=−+ − 552 3 6 56 2 12 12 1 51z2 nnnn7653 n B= Bz() =− z65 3 z + z 4 −+ Sn()=−+−+ 6642 2 2 42 67 2 12 36 42 777 1 nn8777 n 6 nn 42 B=0 Bz () =− z7653 z + z − z + z Sn() =−+ − + 772 2 6 42 78 2 12 24 84

Table 2. Obtaining Sm ( zn, ) from Bzm ( ) .

nn21m+ Bz() SznBznB(,)= () + '() z ++ ... B()m () z m m mm2! m(m + 1)!

B00() z = 1 S(,) zn= n 11n2 B() z=− z S(,) zn=−+ ( z ) n 112 2 2! 11nn23 Bzzz() =22 −+ Sznzz( , )= ( −+ ) n + (2 z − 1) + 2 226 6 2! 3! 3zz2 31zz 22n B() z=−+ z3S( zn , ) =−++ ( z32 ) n (3 z −+ 3 z ) 3322 22 22! nn34 +−(6z 3) + 6 3! 4!

minus a polynomial in z. The operations continue so on until finish. By this operation we observe that we may omit all odd powers terms in z be- ˆ fore and during the transformation of Sz21k+ ( ) into SZk+1 ( ) for k > 0 . From these remarks we may establish Table 3 of transformation. ˆ This algorithm for obtaining SZk +1 ( ) gives rise astonishingly to the easy

calculation of the Faulhaber formula for S21k + ( zn, ) . As examples we have zz22 z Z • Sz( ) = −→ S− ( z) = → SZˆ ( ) = 122 11 2 2 1 S( zn,2) =λλ = (4.8) 1 2 nnn432 z4 1 • Sn( ) =−+→S− ( z) = → SZˆ ( ) = Z2 342432 4 4

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22k+ ˆ Table 3. Change of z in change of Sz21k+ ( ) into SZk+1 ( ) .

2 zZ→ δk 0 zZ42→

6 33 43 2 z→− Z zZ =−3 Z 2

8444 6  443 2 z→− Z z −  zZ =−6 Z + 17 Z 24  10 5 8 6 5 4 3 2 z→− Z10 z − 5 zZ =− 10 Z + 55 Z − 95 Z z12→− Z 615 z 10 − 15 zz 8 − 6 zZ14→− 721 z 12 − 35 zz 10 − 7 8 z16→−−−− Z 828 z 14 70 z 12 28 zz 10 8 zZ18→− 936 z 16 − 126 z14 − 84 z 12− 9z 10

Z 2λλ 14 2 S( zn, ) =+=+Zλλ2 (4.9) 3 2 1! 2 2! nn6555 nn 42 z6 z 4 • Sn( ) =−+ −→S− ( z) =+ 556 2 12 12 6 12

ˆ 1132 SZ3 ( ) = Z − Z 6 12

1132 Sn5 ( ) = u − u 6 12 24λλ23 8 λ SznSZ( , ) =++ˆ′( ) SZˆ ′′ ( ) SZˆ′′′( ) 531! 3 2! 3 3! (4.10) 2Z 14 23 =ZZ − λ +− 2 λλ + 3  63  777 1 • Bz( ) =−+−+ z7653 z z z z 7 2 2 6 42

87 6 42 8 6 4 nn77 n nn− z77 z z Sn77( ) =−+ − +→S( z) =+ − 8 2 12 24 84 8 12 24 ˆ 143 2 77 32 2 SZ4 ( ) =( Z −+6 Z 17 Z) +( Z −3 Z) − Z 8 12 24 11 1 =−+ZZ43 Z 2 8 6 12

1143 1 2 Snuuu7 ( ) =−+ 8 6 12 2λ 4 λ28 λλ 34 16 SznSZ( ,) =ˆ′( ) + SZˆ ′′ ( ) +−(31 Z ) + 3 (4.11) 741! 4 2! 3! 4!

5. Formula for Even Powers Sums S2k ( zn, )

By derivation of S21k + ( zn, ) with respect to z and remarking that

∂=−z Zz21, ∂=z λ n (5.1)

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we obtain the formulae giving S2k ( zn, ) in function of Z,,,λ zn

(21k+) S2k( zn ,) =∂zk S21++( zn ,) = 2( B1( z) ∂+∂Z n2λ ) S 21k( z) (5.2)

j k +1 ( j) (2λ ) =2(Bz1( ) ∂+∂Zk n 21λ )∑ S+ ( Z) (5.3) j=1 j! 12 (23) (22λλ) ( ) ( ) =2BzSZ( )( ) ++ SZ( )  11kk++ 1 1! 2! (5.4) 01 (12) (22λλ) ( ) ( ) +2nSZ( ) ++ SZ( )  kk++11 0! 1!

For examples 1 1 1 • SZˆ ( ) = Z2 , SZˆ′ ( ) = Z, SZˆ′′ ( ) = 2 4 2 2 2 2

3Szn2( ,) = 2 BzSZ 12( ) ′′ ( ) 22λλ ++ nSZnSZ2′( ) 2 2 ′′ ( ) 2 (5.5) =(21z −)λλ ++( Zn 2)

ZZ32 ZZ2 1 • SZˆ ( ) =−=−=−=, SZˆ′ ( ) , SZˆ′′ ( ) Z ,1 SZˆ′′′( ) 36 12 3 2 6 336 ˆ′′ 22ˆ′ˆ ′′ 5Szn4( ,) = 2 BzSZ 13( )( ( ) 22λλ++) 2nSZ( 3( ) + SZ 3( )22λλ +) (5.6)

6. Remarks and Conclusions

The main particularity of this work consists in obtaining Sm ( zn, ) as the trans- m form of z by a differential operator, as so as the Bernoulli polynomial Bzm ( ) ,

from which we deduce the new formula (∂n −∂ zm) S( zn, ) = B m( z) and get

immediately Sm ( zn, ) as polynomials in n. From which we get also the proper- −1 ty saying that B1( zB) 21k + ( z) is a homogeneous polynomial of order k in Z= zz( −1) . The second particularity consists in performing the change of arguments from

z into Z= zz( −1) and n into λ = S1 ( zn, ) in order to get the relation

(∂nz −∂) =Bz12( )( ∂λ −∂Z) which gives rise to the Faulhaber formula of Sm ( zn, ) . From this formula we see that the Faulhaber coefficients are successive deriva- ˆ tives of the function SZSkk++1( ) ≡ 21( z) where Sn21k + ( ) are powers sums on integers. ˆ The relation SZSkk++1( ) ≡ 21( z) leads also to the Faulhaber formula ˆ S21kk++( nSu) ≡ 1( ) where u= nn( −1) . The third particularity of this work is proposing a method for obtaining easily

the Bernoulli numbers Bm from a matrix equation, then of z Bz( ) = mB( z) + B m∫0 mm−1

z Sz( ) = Bz( ) mm∫0

2 (1) n S( zn, ) =++ B( z) n B( z)  m mm2!

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λλ12 ˆ (22) ˆ(2) ( ) S++( zn, ) =++ S′ ( Z) S +( Z) . 21kk1 1! k1 2! ˆ S21kk++( nSu) = 1( )

As conclusion we think that the calculations of powers sums are greatly facili-

tated by the utilization of derivation operators ∂∂zn, and the translation oper-

ator exp(a∂ z ) , parts of Operator Calculus. Operator calculus, which is very different from Heaviside operational calculus, is thus merited to be known and introduced into Mathematical Analysis. More- over it just had a solid foundation and many interesting applications for instance in the domains of Special functions, Differential equations, Fourier, Laplace and other transforms, quantum mechanics [14].

Acknowledgements

The author acknowledges the reviewer for comments leading to the clear proof of the important formula (3.14). The author also acknowledges Mrs. Beverly GUO for her perseverance in the refinement of the article representation. He reiterates his gratitude toward his adorable wife for all the cares she de- voted for him during the long months he performed this work.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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