Oct. 1 5 , 2017.
On Some Sums of Digamma and Polygamma Functions – Revis ion ( 2017 ) and Review
Michael Milgram * ,
Consulting Physicist , Geometrics Unlimited, Ltd.,
Box 1484, Deep River, Ont. Canada. K0J 1P0
Abstract : This paper is an enhanced version of a more than decade - older paper with a similar title. Many formulae involving both finite and infinite sums of digamma and polygamma functions up to quadratic order , few of which appear in standard ref erence works or the literature, but which periodically arise in applications, are collected , reviewed, listed and developed t o the point that a knowledgeable reader could devise a formal proof . Several errors in the literature are corrected.
Author’s Note (2017 version) : Th e forerunner of this paper was ori ginally consigned to arXiv in 200 4 after being rejected by several reputable journals , mostly on the grounds that the contents were well - known , although one referee felt it necessary to opine that the author lacked a “prestigious affiliation”. Over the intervening years, the arXiv entry ( https://arxiv.org/abs/math/0406338 ) has garnered a reasonable number of citations in the published literature , there by demonstrating a need for an easily accessible reference work , and the author has attempted to improve the prestige of his affiliation . Also, s ince th at time, a number of results , particularly for finite sums, have been obtained that may be either not - s o - well - known or new, many of which are needed for applications (e.g. [ 31 ]) . At the same time, new results for Euler (harmonic) sums have independently appeared in the literature, but the correspondence between two equivalent results , one expressed as a sum of (powers of) harmonic numbers, the other as a sum of (powers or products of) digamma or poly gamma functions is not always transparent , recognizable or easily determined . Thus, I have prepared this enhanced version of the original 200 4 paper, containing a large numb er of additional , possibly new, results , intended to be used as a reference or review . As before, the emphasis has been on generalized Euler - type sums – that is, sums where the summand contains at least one digamma or polygamma function, and one or more in dependent parameters, but contains a minimal number (or no) Gamma - functions, except as needed for derivations. With specialized applications in mind, the results are limited to quadratic forms, although a few cubic forms are included when needed for simpli fication. Transcriptions of all the new additions have been checked by copying each formula from the text by hand and verifying numerically using the Maple computer code .
Note that the equation numbers in this and the original version do not always match , although the main text – Sections 1 to 3 – has only been modified slightly . In particular , Appendix B is now devoted to results for finite series . It is interesting to note that some results are closed , rather than consisting of transformations between se ries. A new Appendix C (mostly) repeats the previous listings of infinite series with some additions , again with only hints at the proofs . Appendix D consists of a computer - generated result that was too complicated to transcribe accurately and Appendix E lists a few simple specific and useful results extracted from the more general entries and the literature , regarding which, several corrections are noted . References to now - long - ago - published results , cited previously as unpublished, have also been updated . In general, within
* mike@geometrics - unlimited.com
1 Oct. 1 5 , 2017.
sections, results are listed in order of increasing complexity with only the barest of hints at proofs, except for some cases which are grouped together so that (implied) proofs might become more transparent.
Abstract (200 4 ) : Many formula involving sums of digamma and polygamma functions, few of which appear in standard reference works or the literature, but which periodically arise in applications, are collected and developed. Along the way, a new evaluation for some members o f the family of hypergeometric functions – 4 F 3 (1) – is presented, and a connection is made with Euler sums.
1. Introduction
Although sums of digamma functions often appear in applications such as particle transport [ 19 ] and the evaluation of Feynman diagrams [ 7 , 24 , 25 ] there is a paucity of results available in the standard references [ 12 , 26 ] or in the literature [ 3 , 9 , 13 , 24 , 27 ] except for the special case of Euler or harmonic sums [e.g. 10 ]. Therefore, many formulae are often revisited by different authors [e.g. 27 ] because no repository of quotable results exists. At the same time, many such sums are very slow to converge, so it is important to obtain closed form expressions when numerical evaluation is intended. The purpose of this paper is to collect and develop, in one place, a number of sums involving digamma and polygamma functions that have arisen in appli cations [ e.g. 19 and 31 ] or which were obtained as a byproduct of t h ose endeavour s . Although many of the results about to be quoted are easily established by evaluating parametric derivatives of hypergeometric functions of unit argument - a technique that “traces back to Newton” [ 6 ] - it is believed that many of them, which generalize previously known results, are also new. To keep the paper to a reasonable length, proofs are only sketched , sufficiently that a knowledge able reader , with access to a computer algebra program, should be able to reproduce them.
By way of review, the digamma function ( x ) is defined by d ( x ) log( ( x )) (1) dx so that d ( x ) ( x ) ( x ) ( 2 ) dx with * k 1 ()x ( xkk ), 0. ( 3 ) l 1 (x l )
Polygamma functions are written as:
d d n ()x ()and x (n ) () x ()if xn 2 ( 4 ) dx dx n
* see also ( 7 ).
2 Oct. 1 5 , 2017.
Two useful, well - established results, used throughout, are
k 1 1 ( ( k a 1 ) ( a ) ( k b 1 ) ( b )) /( b a ) ( 5 ) 0 ( a ) ( b ) and 1 1 ( ( a ) ( b )) /( a b ) b a 0 ( a ) ( b ) ( 6 ) ( a ) if b a Also, define generalized harmonic numbers [ 27 , 6 , 35 ]
n 1 Han () ()() naa ( 7 ) l 1 (l a 1) together with n 1 Ha() (( naa )()). n 2 ( 8 ) l 1 (l a 1) I n general, if k 0 , c ommonly used variants are n 1 ( 1) k 1 H(k )(1) ( k 1) ( n 1) ( k 1) (1) n k ( ), ( 9 ) l 1 l ( k )
n 1 ( 1) k 1 H(k )()1 ( k 1) ()() n 1 ( k 1) 1 n 2 1 k ( 2 2 ) ( 10 ) l 1 (l2 ) () k a nd , with respect to the corresponding alternati ng series, n ( 1)l ( 1) k L(k )(1) ( 1) nk [ ( 1) ( n 1) 2 (1 kk ) ( 1) (1 n 1)] (2 (1 kk ) 1) ( 1) (1) n k ( 2 l 1 l ( k ) ( 11 ) (1 k ) n 2 (1 ( 1) )ln(2) 1, k ) .
Throughout, ( n ) is Riemann’s zeta function, 1, k is the Kronecker delta, ( s , c ) is the Hurwitz x l Li( x ) th zeta function , ( 1 ) is the Euler - Mascheroni constant and n n is the n polylog l 1 l function. Symbols sta rting with the letters “ j ” through “ n ” are always non - negative integers. All n 2 other symbols are continuous and (arbitrarily) complex. and 0 if n2 n 1 l 0 l 0 l n 1
(except see ( B. 22 )). G is Catalan’s constant and x symbolizes the “floor” function (greatest integer less than or equal to x).
3 Oct. 1 5 , 2017.
2. Hypergeometric Sums
Lemma 2.1.
An easily obtained and us eful sum needed later (see ( 63 ) ), which generalizes a well - known result [ 18 , Eq. 3.13(43), 26 , Eqs. 7.4.4.40 - 46], which is not listed in any of the standard tables, [ 26 , 34 ] and which is a special case of a more general result [ 26 , Eq. 7.4.4.11 or 7.4.4.3] follows:
1 , n 1 , ( n 1 ) ( ) ( n 1 )( ( n 1 ) ( )) F ( | 1 ) ( 3 2 n 2 , ( ) ( n 1 ) ( 1 2 ) n 1 ( l n ) ) . l 0 ( l n )( l 1 )
Proof: Write the 3 F 2 as a convergent sum with ( ) ( ) , change the lower summation limit from “0” to “ - n”, subtracting equal terms, to get:
1 , n 1 , ( n 1 ) ( ) ( l ) 3 F 2 ( | 1 ) n 2 , ( ) l 0 ( l )( l n 1 ) ( 13 ) ( n 1 ) ( ) ( l n ) n 1 ( l n ) ( ) . ( ) l 0 ( l n )( l 1 ) l 0 ( l n )( l 1 )
The first infinite sum in the second equality is recognized as a 3 F 2 with known summation [ 18 , Eq. 3.13(43)]. ( 1 2 ) follows immediately, for all values of and by the principle of analytic continuation. §
Lemma 2.2
Another useful result needed later (see ( 63 ) ) that does not appear in the standard tables, ( however, see [ 22 ]) but which is a special case of a known result [ 26 , Eq. 7.4.4.11] is
a , b , c ( b ) n ( c 1 ) ( 1 a ) 3 F 2 ( | 1 ) n b , c 1 , ( b c ) n ( c 1 a ) ( 14 ) n 1 ( n a )( 1 ) c ( b n ) ( c b 1 n ) . 0 ( b n a ) ( n ) ( c b n 2 )
Proof: With ( n 1 ) ( ) , apply the well - known result [ 18 , Eq. 3.13(38)] n - times repeatedly and use Gauss’ formula for a 2 F 1 of unit argument. §
Corollary: Evaluate the limit a=1 in ( 9 :
4 Oct. 1 5 , 2017.
1 , b , c c ( ( b ) ( c )) ( n b ) ( b c ) F ( | 1 ) 3 2 n b , c 1 , ( b ) ( b c n )
( 15 ) n 2 ( n 1 )( 1 ) c ( b n ) ( c b 1 n ) , 0 ( b n 1 ) ( n ) ( c b n 2 )
reducing to [ 26 , Eq. 7.4.4(33)] when n=1.
Theorem 2.1
The following terminating sum only appears symbolically in the literature as a limiting case of a very complicated result [ 26 , Eq. 7.10.2(9)], or for special values of “ c ” (e.g. [ 29 , Eq. (3.4)]):
1 , 1 , 1 , k c k ( c l 1 ) F ( | 1 ) ( ( c ) ( ( k 2 ) ( 1 ) ) ) ( 16 a) 4 3 2 , 2 , 1 c ( k 1 ) l 0 ( 1 l )
c k ( l 1 ) ( ( k 2 ) ( ( c k 1 ) ( c ) ) ) . ( 16 b) ( k 1 ) l 0 ( c l )
The two forms of ( 16 ) are obtained from one another using ( 21 ) below . The derivation depends on results to be obtained in Section 4, but this res ult is listed here for consistency of presentation. The proof is outlined in Appendix A.
Theorem 2.2
Similarly, the following, which only appears in the tables as an exceptional, limiting case of a very complicated result [ 26 , Eq. 7.10.2(2)], is easily obtained by rewriting (C. 20 ).
1 1 , 1 , 1 , 1 2 F ( | 1 ) 1 ( 2 ) ( q ) [ ( q ) ][( ( q )] 2 ( 3 ) q 4 3 2 , 2 , 1 q 2 6 ( 17 ) ( 1 ) l ( l 2 ) ( q ) , ( 1 q ) 0 . l 0 ( q l 1 ) ( l 2 )( l 1 )
If q n , n 1 , the infinite series terminates at l n 2 .
Theorem 2.3
Likewise , the following results do not appear in the literature.
5 Oct. 1 5 , 2017.
1 1 , 1 , 1 , 1 2 F ( | 1 ) 7 ( 3 ) 1 ( 2 ) ( 1 n ) ( 1 n ) ( 1 n ) 1 4 3 2 , 2 , 3 / 2 n 2 2 2 2 2 ( 2 n ) 2 ( 18 ) n 1 ( 1 l ) 1 1 ( 1 l ) 1 2 ( l 1 ) ( l n 1 ) l 0 ( l n 2 ) 2 2 1 1 , 1 , 1 , 1 F ( | 1 ) 7 ( 3 ) 1 ( 2 ) ( 1 n ) 2 ( 1 n ) 3 log 2 1 4 3 2 , 2 , 3 / 2 n 2 2 2 2 2 ( 2 n ) 1 1 ( 2 n ) 2 2 log 2 ( n 2 ) ( 19 ) n 1 ( 1 l ) 1 1 ( 1 l ) 1 2 ( l 1 ) ( l n 1 ) l 0 ( l 2 ) 2 2 2 1 , 1 , 1 , 1 7 ( 20 a) 4 F 3 ( | 1 ) ( 3 ) ( 1 log 2 ) 2 , 2 , 1 / 2 4 2 2 1 , 1 , 1 , 1 7 4 F 3 ( | 1 ) ( 3 ) log 2 ( 20 b) 2 , 2 , 3 / 2 4 2
Proof:
1 Set q 2 n in (C. 20 ) employing ( 3 ) and ( C. 23 ) . §
In ( 19 ) , since the series re presentation of the left - hand side does not converge for n 2 , the right - hand side represents the left - hand side * in the sense of analytic continuation, because the variable “ q ” is continuous in (C. 20 ) (whose left - hand side is a valid representation of the hypergeometric function for this range of the variable “ n ”). ( 20 a) is a simple restatement and simplification of ( 19 ) using n= 1. The case n=0, ( 20 b) , correspon ds to the special case a b 0 of [ 15 , Eq.7].
3. Finite sums involving Digamma functions
Theorem 3.1
An important duality exists between two finite sums of digamma functions (see also ( B. 6 ) ) :
k ( b l ) k ( c l 1 ) ( c k 1 ) ( b k 1 ) ( b ) ( c ) . ( 21 ) l 0 c l l 0 b l
Proof: Expand, rearrange , and wi th reference to ( 3 ) and ( 5 ) , find:
1 , 1 , 1 , 1 lim 1 , 1 , 1 , 1 * i.e. F ( | 1 ) means F ( | 1 ) , n 0 . 4 3 2 , 2 , 3 / 2 n q n 4 3 2 , 2 , 3 / 2 q
6 Oct. 1 5 , 2017.
k ( b l ) ( b ) ( b 1 ) ( b 2 ) ( b k ) c l c c 1 c 2 c k l 0 1 1 1 1 ( ( b ) b b 1 b k 2 b k 1 ) /( c k ) 1 1 1 ( ( b ) b b 1 b k 2 ) /( c k 1 ) ( 22 ) ( ( b ) ) /( c ) k 1 ( c k 1 ) ( c 1 l ) ( b )( ( c k 1 ) ( c )) , l 0 b l from which ( 21 ) follows immediately. §
Using ( 3 ) , the sums on either side of the equality can be reduced to one another when “b” and “c” are separated by integers, showing that a closed sum exists for those cases (cf. ( 26 ) and ( B . 3 ) ). Another, equiv alent form of ( 21 ) can be easily found by rearrangement:
k ( b l ) ( c l ) ( ) ( c k 1 ) ( b k 1 ) ( b ) ( c ) c l b l l 0 ( 23 ) ( b k 1 ) ( c k 1 ) ( b ) ( c ) . b c b c
A useful [ 19 ] special case is given below:
k ( l 1 ) k ( l 1 ) 2 ( k 1 ) ( k 2 ) ( 1 ) ( 1 ) 2 2 1 ( 24 ) l 0 l 1 l 1 l 2
Corollary
Apply ( 3 ) to ( 21 ) and simplify to obtain an equivalent form in terms of generalized harmonic numbers [ 6 , 35 ]:
k 1 k 1 H ( 1 ) H ( 1 ) H ( 1 ) H ( 1 ) l 2 1 l k 2 k 1 ( 25 ) l 0 l 1 l 0 l 2 Also see [ 30 , Theorem 7, Example 2].
Theorem 3 .2 (See also (B. 9 ) )
k ( b l ) 1 2 2 2 [ ( b 1 k ) ( b ) ( b 1 k ) ( b ) ] ( 26 ) l 0 ( b l )
Proof: Start from [ 12 , Eq.7.1.1]
7 Oct. 1 5 , 2017.
k ( b l )) ( b 1 k ) ( b ) ( ) /( b a 1 ) ( 27 ) l 0 ( a l ) ( a k ) ( a 1 ) and operate on both sides with giving an interim result worth noting: b
k ( b l ) ( b l ) ( b 1 k ) ( b 1 k ) ( b ) ( b ) ( ) /( b a 1 ) l 0 ( a l ) ( a k ) ( a 1 ) ( 28 ) ( b 1 k ) ( b ) ( ) /( b a 1 ) 2 . ( a k ) ( a 1 )
Then evaluate the limit a b+1 to get ( 26 ) . Alternatively, take the limit c=b in ( 23 ) . §
In applications, the cases b=1 and b=1/2 , both of which are easily obtained from ( 26 ) , arise. * Notably, the case b=1 ext ends an unfinished sum † and integral from the standard tables [ 12 , Eqs. (55.2.1) and (5.13.17)] as follows:
1 1 t k k 1 l 1 log( 1 t ) dt 1 [( ( 1 k ) ( 1 )) 2 ( 1 ) ( 1 k )] 2 0 1 t l 1 l m 1 m 2 ( 29 ) 1 k 1 1 k 1 1 2 2 l 0 ( l 1 ) l 0 ( l 1 ) specializing ‡ a recently quoted general result for non - integral values of “ k” [ 3 , Eqs.(3.6) and (3.9)]: 1 1 t x ( m ) log( 1 t ) dt ( x ) l 1 l 0 1 t l 2 m 1 m ( 30 ) 1 2 2 [( ( 1 x ) ( 1 )) ( 1 ) ( 1 x )]
4. Infinite Series
For completeness, a well - known sum, recently revisited [ 27 , Eq. (2.7)] , is quoted. If ( c b ) 0 ,
( b l ) ( b l ) ( b 1 ) ( b ) [ ( c ) ( c b )] ( ) { ( c b ) } ( 31 ) l 1 ( c l ) l ( c ) b
* Curiously, the computer code Maple 6 (and Maple 2016) found the b=1 version of (the new result) ( 26 ) , but missed all other possibilities tested. Mathematica [ 34 ,Version 4.1 (~2004)] missed all possibilities tested, but Mathematica [ 10 , version(2016)] correctly found this result. † reproducing a known result [ 1 , Lemma 1] in this special case. ‡ Note the relationship between t he inner sum in ( 30 ) and ( 67 ).
8 Oct. 1 5 , 2017.
Set c=b+1, and change the notation slightly, giving
( b l 1 ) 1 2 { ( b )[ ( b 1 ) ( 1 )]} . ( 32 ) l 0 ( l 1 ) ( b l 1 ) b 6
In the limit b 0 this reduces to a very well - known result [ 9 , 24 , 34 ]
( l 1 ) 2
2 ( 3 ) ( 33 ) l 0 ( l 1 ) 6 Theorem 4. 1
Consider a useful variant of ( 32 ) :
( l 1 ) 1 2 2 { ( p ) ( p ) ( q ) ( q ) } ( 34 ) l 0 ( l p )( l q ) 2 ( q p )
Proof: Temporarily, assume ( ) 1 , and consider
( l ) 1 1 ( l ) 1 1 [ ] l 0 ( l 1 ) ( l p )( l q ) ( q p ) l 0 ( l 1 ) ( l p ) ( l q ) 1 ( ) ( p ) , p ( ) ( q ) , q [ F ( 1 ) F ( 1 ) ] ( q p ) ( p 1 ) 2 1 p 1 ( q 1 ) 2 1 q 1 ( 35 ) ( ) ( 1 ) ( p ) ( q ) ( ) ( q p ) ( p 1 ) ( q 1 ) using Gauss’ summation for a 2 F 1 of unit argument. The condition on may now be relaxed. Operate on both sides of ( 35 ) with lim , respective ly reducing the left - and right - hand sides 1 to the corresponding sides of ( 34 ) . §
The sums (enclosed in square brackets) in these results diverge if they are considered individually w hen ( ) 1 , so that a careful adherence to the ordering of limits, and the use of the principle of analytic continuation is needed.
Corollary
The following generalizes ( 33 )
( l 1 ) ( q ) ( q ) 1 ( 2 ) ( q ) 2 2 ( 36 ) l 0 ( l q )
Proof: Evaluate the limit p q in ( 34 ) . § See also ( 66 ) and ( .
9 Oct. 1 5 , 2017.
Corollary
( l m 1 ) 1 { ( ( p ) ( p ) 2 ( q ) ( q ) 2 ) / 2 l 0 ( l p )( l q ) ( q p ) ( 37 ) m 1 ( l 1 ) ( p ) 2 ( q ) 2 ( p ) ( p m ) ( q ) ( q m ) } l 0 ( p l 1 )( q l 1 )
Proof: Using ( 3 ) and ( 6 ) assume p , q 1 , 2 , m and split the following sum using partial fractions
m 1 1 k 0 l 0 ( l p )( l q )( l m k ) ( 38 ) ( p ) m 1 1 ( q ) m 1 1 m 1 ( m l ) ( q p ) l 0 p m l ( q p ) l 0 q m l l 0 ( q m l )( p m l )
The first two sums are easily evaluated; reverse the third right - side sum, apply ( 34 ) to the finite sum on the left and eventually arrive at ( 37 ) . §
The conditions on p and q can be relaxed by taking the appropriate limits. A limit must also be evaluated when p and q are separated by integers. In particular, many results, laboriously and individually derived elsewhere [ 24 ] are special cases of ( 37 ) . Also, if p and q are integers, the finite sum in ( 37 ) can be evaluated in close d form using ( 26 ) – also see ( B . 3 ) , ( B. 16 ), ( C. 6 ) ) and (C. 7 ) as well as Xu [ 36 ] ) .
Theo rem 4. 2
[ ( q l ) ( l )] ( , q ) where l 0 l 1 1 ( 39 ) ( , q ) ( ( q 1 ) 2 ( 1 ) 2 ( q 1 ) ( 1 ) ) 2 ( 1 )( ( q 1 ) ( 1 ))
Proof: Consider the following sum with ( ) ( ) :
(qll )()1() q q ,1,1 () b ,1,1 [ ] 32F( |1) 32 F ( |1) l 0 (ql )()(1)() ll q q ,2 () a ,2 ( 40 ) (x 1) ( 1) ( 1) ( q 1) ( q 1) ( 1) ( b ) ( ) (x 1) ( 1) ( 1) ( q 1)
10 Oct. 1 5 , 2017. using a well - known result [ 18 , Eq. 3.13(43)] to sum the hypergeometric functions. Relax the constraint and operate on both sides of ( 40 with lim so that the right - and left - hand sides become the corresponding sides of ( 39 ) . §
As before, the two terms in ( 40 ) correspond to divergent series if written individually; this is indicated by the use of square brackets as a reminder, and again new results can be obtained by operating with and/or and taking appropriate limits. For example, the limit 1 in q ( 39 ) yields
( q l 1 ) ( l 1 ) 2 1 ( ( ( q ) ( 1 )) ( 2 ) ( q ) ) / 2 ( 41 ) l 0 ( l 1 )
and operating with on ( 41 ) with some reordering gives q
( q l ) 1 ( 2 ) ( q 1 ) ( q 1 ) ( q 1 ) ( 1 ) ( q 1 ) . ( 42 ) l 0 ( l 1 ) 2
Take the limit q=1 to recover the special case
( 1 l ) 2 ( 3 ) , ( 43 ) l 0 ( l 1 ) corresponding to a known result [ 24 ], [ 12 , Eq. 55.9.7).
Theorem 4. 3 Eq. ( 39 ) can be generalized further.
[ ( q l ) ( l )] m 1 ( q l m ) ( l m ) ( m , q ) ( 44 ) l 0 l m 1 l 0 l 1
Proof: Add and subtract terms corresponding to l m , , 1 in ( 39 ) and re - order the resulting series. §
1 1 Corollary : ( 44 ) reduces to the following useful result by setting m 2 and q 2 .
11 Oct. 1 5 , 2017.
[ ( l m 1 ) ( l m 1 )] 2 1 1 2 2 2 ( ( 1 ) ( 2 ) ( m 1 ) ( m 1 ) ) l 0 ( l m 1 )
m 1 1 ( 45 ) 1 1 ( l 2 ) 2 ( ( 1 ) ( 2 ) ) ( 2 ) ( 1 ) l 0 ( l 1 ) See also ( 24 ).
Theorem 4. 4
For completeness’ sake, a known variation of ( 31 ) [ 27 , Eq. (2.14)] is quoted, without proof.
( q l ) ( q ) ( p q 1 ) ( l 1 ) ( ( 1 ) ( p q 1 ) ( p 1 ) ) ( 46 ) l 0 ( p l ) ( p 1 ) ( p q )
Using methods similar to those already employed (e.g. T heorem 4.2), other variants of ( 31 ) and ( 46 ) can be obtained:
( q l ) ( p l ) ( q 1 ) [ ( p q ) ( p 1 ) l 0 ( p l )( l 1 ) ( p 1 ) ( 47 ) ( p 1 )[ ( p 1 ) ( p q )] ]
and
( q l ) ( 1 l ) ( q 1 ) [ [ ( p 1 ) ( p q )] 2 l 0 ( p l )( 1 l ) 2 ( p 1 ) ( 48 ) 2 [ ( p 1 ) ( p q )] ( p q ) ( p 1 ) ]
Theorem 4. 5
1 ( c l ) ( c ) ( ( f 1 ) ( p c 1 ) ) ( f ) ( 1 f ) ( 1 c ) ( f l ) { ( p l ) ( p c 1 ) ( p 1 ) ( f c ) ( p f ) l 0 ( 49 ) ( 2 f ) ( p c ) ( f c l )( 1 ) l } ( p f ) ( f c ) l 0 ( 2 c l ) ( l 1 ) ( p 1 l )( 1 c l )
Proof: Consider the sum
( c l ) ( l ) ( c ) ( ) 1 , , c F ( | 1 ) ( 50 ) 3 2 p , f l 0 ( p l ) ( f l ) ( p ) ( f )
Convert the hypergeometric 3 F 2 into another 3 F 2 having the property that it terminates if p m , ( m >1) according to [ 18 , section 3.13.3]
12 Oct. 1 5 , 2017.
1 , , c ( p f 1 c ) ( p ) ( 1 ) ( f ) ( 1 c ) 3 F 2 ( | 1 ) { p , f ( p ) ( f c ) ( p c ) ( f ) ( 51 ) ( f 1 ) F ( f c , 2 p , 1 c | 1 ) } ( 1 c ) ( f c 1 ) ( p 1 ) 3 2 f c 1 , 2 c
lim Operate on both sides with and simplify. § f
This derivation of ( 49 ) is valid for all continuous values of p , although it is not particularly useful
unless the infinite series (equivalent to a 3 F 2 ) can be summed analytically (e.g. p m , an integer, in which case the infinite series terminates). If f =1, it can be shown, by summing the infinite series [ 26 , Eq.7.4.4.(49)] and taking the app ropriate limit, that ( 49 ) reduces to ( 46 ) as a special case. lim To obtain other useful results from ( 49 ) , operate with . Then set p 2 giving f 1 f ( l q ) ( l 1 ) 1 1 2 1 2 1 2 ( q 1 ) ( 2 ( 1 q ) 2 ( 1 q ) 4 2 ( 1 q ) ) , ( 52 ) l 0 ( l 2 )
reducing to ( 43 ) when q =1.
Alternatively, operate on ( 49 ) with lim lim , and from the requirement that the coefficient p c 1 f 1 f 1 of ( p c 1 ) on the right - hand side must vanish (series converges), find
( l q ) ( l q ) ( q ) 2 ( q ) cot q , 2 ( 53 ) l 0 ( l 1 )( q l ) sin q 6
reducing to ( 33 ) when q 1 . See also (C. 19 ) to ( C. 23 ) .
Lemma 4. 6
The following sums the difference of two divergent series (see also ( C. 43 ) and ( C. 44 )).
( l 1 ) ( l 1 ) 2 2 1 ( 54 ) l 0 l 2 3
Proof: Consider the following sum (with ( ) 0 ):
13 Oct. 1 5 , 2017.
1 1 ( l 2 ) ( l 1 ) ( l ) ( l 2 ) 3 l 0 ( l ) ( l 1 ) 2 ( 55 ) ( 1 ) ( ) ( 1 ) 2 F ( 1 , 1 / 2 | 1 ) 2 F ( , 1 / 2 | 1 ) . 3 2 1 3 / 2 3 2 1 3 / 2 ( 2 ) ( 2 )
As before, sum the hypergeometric series, relax the limitation on and, with deference to the lim principle of analytic continuation, operate on both sides with . § 1 Theorem 4. 7
The following generalizes ( 54 ) (also see (C. 9 ) etc . ):
(lk1 ) ( lk 1) [2 ] ()(kk 1 )(()4)(() 1 kk ( 1 )) 1 2 2 2 l 0 l 2 ( 56 ) 2 k 1 (l ) ()12 8 4() 1 2 2 2 1 6l 1 (l 2 )
Proof: Let
( l k 1 ) ( l k 1 ) V 2 k 1 ( 57 ) l 0 l 2 Then 1 1 1 ( 1 ) ( k ) ( 1 ) ( k 1 ) V V ( )( ) V 2 2 2 k k 1 1 1 k 1 1 ( 58 ) l 0 l k 2 l k l 2 k 2 k 1
Apply ( 58 ) repeatedly (k times), giving
k ( 1 ) ( l ) ( 1 ) ( l 1 ) 2 V V 2 2 2 with V k 0 ( 1 ) 0 ( 59 ) l 1 l 2 l 1 3
2 The second term enclosed by brackets in ( 59 ) equals / 2 if l 1 . Use ( 54 ) and its analogues, and, after some simplification, including the use of ( 21 ) , find ( 56 ) . § See also ( 45 ).
Theorem 4. 8
( a l ) ( a l q ) 2 2 [ ] ({ ( a q ) ( a q ) } { ( a ) ( a ) }) / 2 ( 60 ) l 0 ( a l ) ( a l q )
Proof: Since this is effectively the (finite) difference between diverging sums, consider the following two converging sums, subject to the condition ( a ) ( ) :
14 Oct. 1 5 , 2017.
( l ) ( l q ) [ ] l 0 ( a l 1 )) l 0 ( a l q 1 ) ( 61 ) ( ) ( q ) F ( , 1 | 1 ) F ( q , 1 | 1 ) ( a 1 ) 2 1 a 1 ( a q 1 ) 2 1 a q 1
As in previous cases, evaluate the two hypergeometric functions independently, simplify, and operate with . Then, unite the two sums into one (converge nt) sum, and evaluate the limit a . §
Notice that, to within a constant, each of the terms of ( 60 ) that are enclosed in braces ( { } ) regularize corresponding divergent series on the left . Also, see (C. 8 ) and (C. 14 ) .
Corollary
( q l ) ( q l ) 1 ( 2 ) ( q ) ( q ) ( q ) 2 2 ( 62 ) l 0 ( q l ) ( q l )
lim Proof : Operate with on ( 60 ) ; alternatively, operate with lim lim on ( 26 ) . § a 0 q b q k b
Theorem 4. 9
( c l ) ( n l 1 ) 2 1 [ ] ( n 1 ) ( ( c ) ) l 0 ( n l 1 ) ( c l ) 3 ( c n 1 ) ( 63 ) ( c ) n ( l 1 ) ( c n 1 ) ( c n 1 ) 2 2 ( c n 1 ) l 0 ( c l n 1 )
Proof: Consider the following sum with ( c ) 0 . With this restriction, the sum may be split into two and identified:
( l ) ( l n 1 c ) [ ] l 0 ( c l )( n 1 l ) ( n 1 l )( c l ) ( 64 ) ( ) 1 , n 1 , ( n 1 c ) 1 , c , n 1 c F ( | 1 ) F ( | 1 ) ( n 1 ) ( c ) 3 2 c , n 2 ( c ) ( n 1 ) 3 2 n 1 , c 1
Using ( 1 2 ) and ( 9 , each of the hypergeometric functions can be summed, and the restriction relaxed by the principle of a nalytic continuation. Since the combined sums converge, operate on lim both sides with and sum a terminating F , giving c 3 2
15 Oct. 1 5 , 2017.
( c l ) ( n l 1 ) 2 [ ] ( c 1 n ) ( n 1 ) ( ( c ) ( c n ) ) ( n l 1 ) ( c l ) 3 l 0 n 1 ( 65 ) 2 ( c n ) ( c l n ) n 1 n , 1 , 1 , 1 ( 1 , c n ) ( c n ) 4 F 3 ( | 1 ) ( c n 1 ) l 0 ( l 1 ) ( c n ) c 1 n , 2 , 2
From either version of ( 16 ), the 4 F 3 can be identified, and Eq. ( 63 ) emerges after considerable simplification. § See also ( B. 16 ).
In the limit c n 1 , both sides of ( 63 ) and ( 65 ) vanish. This has been verified and a known, closed - form for a special case of the resulting 4 F 3 has been retrieved.
Theorem 4.10 (generalized Euler sums [ 10 ] and polylogarithms [ 16 ]. See also (C. 14 ) .
n n 2 ( l 1 ) ( 1 ) ( 1 k ) ( n k 2 ) n 2 ( n ) ( q ) ( q ) 1 ( q ) , n 2 ( 66 ) n k 2 l 0 ( l q ) ( n 1 )! k 0
n 2 Proof : Operate on ( 36 ) with . § dq n 2 (n ) n 1 Using (1) ( 1)n ! ( n 1) and setting q=1 gives a simple derivation for an equivalent form of Euler’s (linear) sum:
( l 2 ) ( 1 ) n n 2 ( 1 ) ( n 1 ) 1 k ( k 1 ) ( n k ) n ( n 1 ) ( 67 ) l 0 ( l 1 ) 2 k 1 w hich, when compared to the form usually quoted in the literature [ 1 , Eq. 20], [ 10 , Theorem 2.2] , yields the interesting relation n2 n 2 (k 1) ( nk )2 kk ( 1) ( nk ), (n 1) ( 67 a) k1 k 1 a result that is true for any indexed function (proof by reversing the right - hand sum). Xu [ 36 , Eq s . (1.27) and (1.28) – incorrect – add a term 1/ a s 1 to the latter) has recently obtained an equivalent form of ( 66) as have Sofo and Cvijovo ́ [ 28 , Theorem 1]. Special cases of ( 66) with 1 combinations of q0,2 ,1and n 2,3 can be found in Appendix E; for example, s et n=3 in ( 66 ) to retrieve the well - known harmonic result [ 9 , Eq. (17)]: (l 1) (1) 4 . 3 ( 68 ) l 0 (l 1) 360 Finally, by taking the limit q m in ( 66 ) using n 2 we find
(1 l ) 2 (2) (m 1)(1 ( m 1)) ( m 1) . 2 3 2 ( 69 ) l 1 (l m ) m l m
16 Oct. 1 5 , 2017.
Split the sum into its finite and infinite parts, apply the limit q 0 to ( C. 6 ) and compare the resulting infinite sums to obtain ( B. 68 ). lim Variations of Euler (quadratic) sums are available by operating on ( 47 ) with : q 1 q (ql ) ( ql 1) 1 2 ( [()qq ( 1)] ()( qq 1)[() q ] l 0 (ql )(1 l ) ( q 1) 6 ( 70 ) 2 (3) (q 1) ( q ) ) reducing – after considerable simplification - to a known result [ 9 , Eq.(9)], [ 16 , Eq.(8.92)] when q 1 : [ ( 1 l ) ( 1 )] 2 11 4 2 ( 71 ) l 0 ( 1 l ) 360 Notice that the left - hand side of ( 70 ) can be rewritten as a transformation by a simple single step recursion, that is ()(qlql 1) () ql 2 () ql 2 , ( 70 a) l0(qll )(1 ) l 0 ( qll )(1 ) l 0 ( ql ) (1 l )
and so, in the limit q 1 (1l )2 (1 l ) 22 4 2 3 3 (3) ( 72 ) l0(1l ) l 0 (1 l ) 6 30 equivalent to Euler sums of weight (2,2) and (1,3), both of which are known ( e.g. [ 5 ], [ 9 ] ) .
Alternatively, simple differentiation of ( 47 ) with respect to the variable q with ( p>q+1 ) produces (ql )( plql )( ) (1) q [ (q 1){ ( pq ) ( p 1) l 0 (pll )(1 ) ( p 1) ( 73 ) (p 1)( ( p 1) ( pq ))} (p 1) ( pq ) (2) ( pq )] , lim reducing to ( 70 ) when p=q+1 . Similarly, operating on ( 48 ) with gives p q 1 p q ()(qlql 1)(1)1 l 5 2 3 2 ( ()q4 ()( q 2 () q ) l 0 (q l )(1 l ) 2 12 2 2 1 11 4 2 (qq )((2) ( ) 4 (3)) ( qq )) / ( 1) 2 180 6 ( 74 ) 1 32 2 1 (()qqq3 ()( () ) (2) () q (3)/( ) q 1) 2 2 2 422
In the limit q= 1 , a straightforward result is
17 Oct. 1 5 , 2017.
( 2 l ) ( 1 l ) 2 1 7 1 3 2 4 ( 3 ) 2 4 ( 3 ) 2 6 ( 5 ) 2 6 72 3 ( 75 ) l 0 ( 1 l )
Using this technique, along with ( 3 ) , variations in the choice of lim and partial fraction p q n decomposition, all Euler sums up to cubic order become accessible. The use of computer algebra is recommended.
Theorem 4.11
2 2 ( c l ) ( 1 l ) ( c ) 1 , 1 , 1 , p c ( c ) 2 4 F 3 ( | 1 ) [ l 0 ( p l ) ( p ) 2 , 2 , p ( p 1 )( p 1 c ) 6 ( 76 ) ( p 1 c )[ 2 ( p 1 ) ( c )] ( p 1 c ) ( p 1 c ) 2 ( p 1 )[ 2 ( c )] ] and
()(1)()cllc 1,1,1, p c () c F( |1) ( ( p 1) 4 3 2, 2, p l 0 ()pl () p (1)(1) ppc ( 77 ) [ (p 1) ( pcp 1 )][ ( 1) ( c )] )
Proof:
Following the method of Theorem 4.5, operate on both sides of ( 50 ) and ( 51 ) with both lim lim 2 lim lim 2 and . The resulting expressions contain a large number of f 1 f 2 1 f f 2 terms, all of which can be evaluated (with computer algebra) using results given elsewhere in this paper, especially ( 47 ) and ( 48 ) . Add and subtract the resulting expressions. § For the case p c 1 , see (C. 20 ) . Note: – the version 2014 statement of ( 77 ) contained a transcription error.
Seen from another viewpoint , subtracting ( 76 ) from ( 77 ) gives an alternative identification for a particular 4F 3 (1) specifically
1,1,1, c ()p ( pcl ) 2 4F 3 ( |1) ( (1 l ) (1 l ) ) 2, 2, p 2(pc )l 0 ( pl ) ( 78 ) (p 1) ( 2( (pc )) 2 2 /6 ), pc 1and c 1, 2(c 1)
w here (c 1) ( p 1) after the substitution c p c , the significance of which will be discussed elsewhere ( to be published ) . See also ( 16 ) and ( B. 43 ) .
18 Oct. 1 5 , 2017.
Acknowledgements
Although I had been thinking about updating this paper for some time, I am grateful to Prof. Lu Wei for motivating me to finally do so. I am also grateful to those (cited) authors who chose to make copies of their work freely and publicly accessible , and acknowledge the Researchgate computer algorithm which brought many of those papers to my attention. Some authors have chosen not to post freely available preprints or legall y permitted offprints on public websites; citations to any such papers published after 2005 and only available behind a paywall, have been withheld. Please contact this author for further information .
References
1 . Adamchik V., “On Stirling numbers and Euler Sums”, J. Comp. Appl. Math.,79,119 - 130 (1997).
2 . Adegoke, K and Layeni, O., “New Finite and Infinite Summation Identities Involving the Generalized Harmonic Numbers”, https://arxiv.org/abs/1508.07345 (2015).
3 . Borwein, J.M, Zucker, I.J and Boersma, J., “The ev aluation of character Euler double sums” (2004). available from: http://eprints.cecm.sfu.ca/archive/00000255/
4 . Choi, J . , “Finite Summation Formulas Involvi ng Binomial Coefficients, Harmonic Numbers and Generalized Harmonic Numbers ”, Journal of Inequalities and Applications, 2013:49 (2013).
5 . Choi, J . , “ S ummation F ormu las Involving B in omial C oefficients, Harmonic Numbers and G eneralized Harmonic N umbers”, Abstract and Applied Analysis , Article Id 501906 (2014). http://dx.doi.org/10.1155/2014/501906.
6 . Chu, W. and De Donno, L., “Hypergeometric Series and Harmonic Number Identities”, h ttp://www.arXiv.org: math CO/0403353, (2004).
7 . Coffey, M.W., “ On one - dimensional digamma and polygamma series related to the evaluation of Feynman diagrams”, J. Computational and Applied Mathematics, 183 (2005) 84 - 100. doi:10.1016/j.cam.2005.01.003. Also available from https://arxiv.org/abs/math - ph/0505051 .
8 . Cvijovic, D., “Closed - form summations of certain hypergeometric - type series containing the digamma function”, J.Phys. A: Math. Theor. 41, 1 - 7, (2008) . Doi:10.1088/1751 - 8113/41/45/455205. Also available from https://www.researchgate.net/publication/231110 852.
2 9 . de Doelder, P.J., “On some series containing ( x ) ( y ) and ( ( x ) ( y )) for certain values of x and y ”, J. Comp and Appl. Math., 37, 125 - 141 (1991).
10 . Flajolet, P. and Salvy, B., “Euler Sums and Co n tour Integral Representations”, Experimental Mathematics 7,1,15 - 35 (1997).
19 Oct. 1 5 , 2017.
11 . Glasser, Larry, (private communication), (2017).
12 . Hansen, E.R, “A Table of Series and Products”, Prentice - Hall (1975).
13 . Jord an, P.F., “Infinite Sums of Psi Functions”, Bull. American Math. Soc., 79,4,681 - 683 (1973).
14 . K ö lbig, K.S., “The polygamma function ψ (k) (x) for x=¼ and x=¾”, J. of Comp. and Appl. Math., 75 (1996) 43 - 46. Doi: 10.1016/S0377 - 0427(96)00055 - 6
15 . Lavoie, J.L., “Some Summation Formulas for the Series 3F2(1)”, Math. Comp., 49,179,269 - 274 (1987).
16 . Lewin, L., “Polylogarithms and Associated Functions”, North Holland, (1981).
17 . Liu H., Zhou, W. and Ding, S. “Generalized harmonic number summation formulae via hypergeometric series a nd digamma functions”, Journal of Difference Equations and Applications” , 1 - 15, (2017). dx.doi.org/ 10.1080/10236198.2017.1318861 citation withheld (see acknowledgements)
18 . Luke, Y. , “The Special Functions and their Approximations, Vol. 1”, Academic Press (1969).
19 . Milgram, M.S., “Identification and Properties of the Fundamental Expansion Functions for Neutron Transport in an Infinite Homogeneo us Scattering Medium”, Annals of Nuclear Energy, 32,11, 1167 - 1190, (2005). https://doi.org/10.1016/j.anucene.2005.02.015 . Also a vailable from https://www.researchgate.net/publication/239358548 .
20 . Milgram, M.S., “A Series Expansion for Integral Powers of Arctangent”, Integral Transforms and Special Functions, 17, 7, 531 - 538 ( 2006); also avail able from http://www.arXiv.org : math:CA/0406337, (2004).
21 . Milgram, M.S., “Powers of a Hypergeometric Function”, Integral Transforms and Special Functions, 17,11, 829 - 832 (2006) ; also available from https://arxiv.org/abs/math/0505202
22 . Milgram, M. , “ Comment on a paper of Rao et al. , an entry of Ramanujan and a new 3F2(1)”, J. of Computational and Applied Mathematics,201,1 - 2 (2007). http s ://d oi .org/ 10.1016/j.cam.2006.01.025. Also available from h ttps://www.researchgate.net/publication/242980686
23 . Milgram, M.S., “4 47 Instances of Hypergeometric 3 F2(1)” , available from https://arxiv.org/abs/1105.3126 (2011) and/or https://www.researchgate.net /publication/51891949_447 .
24 . Ogreid, O.M. and Osland, P.O., “More Series related to the Euler Series”, J. Comp. & Appl. Math.,136, 389 - 403 (2001).
20 Oct. 1 5 , 2017.
25 . Ogreid, O.M. and Osland, P.O.,”Some Infinite Series Related to Feynman Diagrams”, J. Comp. & Appl. Math, 140, 659 - 671 (2002).
26 . Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I., “Integrals and Series”, Vol.3, “More Special Functions”, Gordon and Breach, 1990.
27 . Rassias, T.M and Srivastava, H.M., “Some Classes of Infinite Series associated with the Riemann Zeta and Polygamma functions and G eneralized H armonic N umbers”, Appl. Math. & Computation, 131, 593 - 605 (2002).
28 . Sofo, A. and Cvijovi ́, “Extensions of Euler Harmonic Sums” , Appl. Anal. Discrete Math., 6 (317 - 328 (212). Available online at http://pefmath.etf.rs
29 . Srivastava , H. M., Studies in Appl. Math., 86, pp . 79 - 86 (1992).
30 . Speiss, J. , ”Some Identities Involving Harmonic Numbers”, Mathematics of Computation, 55, 192, 839 - 863 (1990).
31 . Wei, Lu , “A Proof of Vivo - Pato - Oshanin’s Conjecture on the Fluctuation of von Neumann Entropy” , Phys. Rev. E , 96 , 022106 (2017). Also available from ar X iv.org/abs/1706.08199 (June 26, 2017).
32 . Wei,C., Yan,Q. and Gong,D ., “A family of summation formulae involving harmonic numbers”, Integral Transforms Spec. Funct. 26(2015) 667 - 677. http://dx.doi.org/10.1080/10652469.2015.1034124 citation withheld ( see acknowledgements )
33 . Wei, C., “Watson - type 3 F 2 - Series and Summation Formulae Involving Generalized Harmonic Numbers”, ar X iv.org/abs/1607.01006 (July 3, 2016).
34 . Wolfram, (2004): http://www.functions.wolfram.com /
35 . Wu, T.C., Tu, S - T, Srivastava, H.M., “Some Combinatorial Series Identities Associated with the Digamma Function and Harmonic Numbers”, Appl. Math. Lett., 101 - 106 (2000).
36 . Xu Ce, “Parametric Harmonic Number Identities” , ar X iv.org/abs/1701.03726 (2017) .
37 . Xu C.,Yang, Y. and Zhang, J.,” Explicit Evaluation of quadratic Euler Sums”, International Journal of Number Theory,(2017) doi: 10.1142/S1793042117500336. Also available from https://arxiv.org/abs/1609.04923 .
38 . Zheng,D.,”Further summation formulae related to generalized harmonic numbers”, J.Math. Anal. Appl., 335 692 - 706, (2007). doi:10.1016/j.jmaa.2007.02.002 .
21 Oct. 1 5 , 2017.
Appendix A : Sketch of Proof of ( 16 ) *
Proof: In ( 65 ) , with ( c ) 0 , set n=0, giving
()(1)cll 2 () c [ ] (1)()()c c c 2 (A . 1) l 0 (lcl 1) ( ) 3 ( c 1)
Split the infinite sum into two parts: l 0 k , , and l k 1 , and shift the lower value of the summation index of the second (infinite) series to zero. The resulting infinite serie s, given by ( 65 ) with the replacement c c k 1 and n k 1 , involves a 4 F 3 , so using (A . 1) and ( 21 ) , solve for the 4 F 3 and simplify. §
Appendix B : Finite Sums
The following useful and related finite sums, based on the methods of this paper, many of which are needed [ e.g. 31 ]) or obtained elsewhere [ 19 ], are presented with minimal proof.
From [ 21 , Eq.(3)]
k ( l 1 ) k ( l 1 ) 2 2 2 ( 1 ) H ( 1 ) 2 H ( 1 ) 1 2 k 2 k ( B . 1 ) l 1 ( k l 2 ) l 1 l
From ( B . 5 ) (below) , after some rearra ngement and utilizing ( 26 ) obtain
k (l 1) k (l 1 ) 2 1H() 12 H () 1 1 2 k1 2 k 1 2 l0(k l 2 ) l 0 l 1 ( B . 2 ) 3 1 Hk1(1) ( k 2 ) 2 ( 2) H k 1 ( 2 ) and k ( l 1 ) 2 ( k 2 ) ( k 2 ) ( k 2 ) 2 ( B . 3 ) l 0 ( k l 1 ) 6 which, after reversal, or with reference to ( 21 ) , can also be written
k (k l 1) 2 (k 1) ( k 1) ( k 1)2 . l 1 l 6
Also , s ee the case n in ( B. 53 ) below. Equivalently,
* A proof of ( 16 ) based on [ 26 , Eq. 7.4.4(38)] might be attainable. However, that result fails several numerical tests, and appears to be incorrect.
22 Oct. 1 5 , 2017.
k1 l 1 kl 11 1 1 l0(kl2 )(1) m 0 m lm 00 (1)( l m 2 ) ( B . 4 ) 112 1 1 22 Hk2( ) H k 2 ( 2 ) HH kk 12 (1) ( 2 ) and f rom [ 21 ] (also see (B. 13 ))
k1l 1 k 1 l 1 2 . ( B . 5 ) l0klc m 0 mc lm 00 lc 2 mc
n Follow the same steps as in ( 22 ) and operate with to generalize ( 21 ) : b n
k (n ) (b l ) ()n()(,)bsc () n ( bk 1)(, sck 1) (c l ) s l 0 (B. 6 ) k (,s c l 1) (1)! n n , n 1 l 0 (b l ) where s is arbitrary , except ≠ 1 . In the case = 1 , = 1 , differentiate ( 21 ) once with respect to the parameter b to find
k()()blcl k (1)() bkb (ck 1) ( bk 1) 2 l0clbl l 0 () () bc (B. 7 ) (ck 1) ( bk 1) () b () c ()()b c . (b c ) 2 In the limit → , we obtain
k ()bl k ()1 bl (2) (bk 1) (1)(1) b kb k bl ( bl )2 2 l0 l 0 (B. 8 ) 1 ()()b b (2) (). b 2 Further, i n ( B. 6 ) set b c , s n 1 and solve, thereby producing a variant of ( 26 ) :
k ( n ) ( b l ) ( 1 ) n n ! { [ ( 2 n 1 ) ( b 1 k ) ( 2 n 1 ) ( b )] ( b l ) n 1 2 ( 2 n 1 )! l 0 (B. 9 ) 1 [ ( n ) ( b 1 k ) 2 ( n ) ( b ) 2 ] } , n !
Not ing that ( 21 ) is valid for all c , let c → − − 1 , evaluate that limit, compare the coefficients of t he first order expansion terms and replace c → to find
23 Oct. 1 5 , 2017.
k(bl ) 2 k ( kl 1) 2 (bk 1) ( k 2) ( b ) . (B. 10 ) l0(kl 1) 6 l 0 bl
See also ( B. 7 ). An equivalent form of ( B. 10 ) can be found by reversing each of the sums, and replacing → − to yield
k(b l ) 2 k ( l 1) 2 (b 1) ( k 2) ( bk ) . (B. 11 ) l0(l 1) 6 l 0 b l
The case (B. 9 ) can also be used to find more general results. For example, after partial fraction decomposition and some rearrangement, the case n= 1 using ( B. 8 ) and (B. 9 ) gives
k 1 (1)l (46 kkkk4 3 2 42) ()k2 () k (1ll )22 (2 ) kk 22 ( 1) l 0 . (B. 12 ) (3)()k (5 k 1)( k 1) 7 42 6k 2 1806
Also, simplifying [ 21 , Eqs. (3) and (5)] yields
k k (1)lc1 (1) lc 1 1 2 (B. 13 ) 2 ((2ck 1) (2) c 2 ( kc 1) ) () c 2 (). c l0(2)cl l 0 ( klc )
In the case that = − , an extensive series of limits utilizing ( 26 ) and (B. 44 ) among others , eventually reduces (B. 13 ) to ( B. 10 ). Note that ( B. 16 ) below is a special case of (B. 13 ) , both in the limit → 0 . Also see ( B. 66 ).
S everal new results can be obtained by comparing ( 16 ) with [ 31 , Eq. (A.24) ( generalizing → ) ] to obtain
m(cl 1) m ( cml ) 1 2 ( (1))m 2 ( ) (B. 14 ) l1l l 1 l
w here (c m ) ( c 1) . Further, reversing either term in the left - hand side of ( B. 14 ) identifies the related sums
mm(1)(cl cml ) m ( cml ) m (1) cl (m 1) ( m 1) (B. 15 ) ll11l l l 1 lml(1) l 1 lml (1)
and a similar result exists for ( B. 16 ) below , if the summations are combined . A fter applying ( 19 ) to ( B. 14 ) we find another variation
24 Oct. 1 5 , 2017.
m ()()1l l 2 1 ( ) ( (cm 1) ( c 1) ) ( ( cm 1) ( c 1)) l 1 cl1 cml 2 2 ( B. 16 )
The special case = / 2 in ( B. 16 ) requires a careful evaluation of limits . To first order, t he case = 2 − 1 , yields
2j 1 (l j 1 ) 2 2 1 1 1 (2j )[ (2)] j 2 ( 2 j ) , (B. 17 ) l 1 l 4 which can also be written ( by splitting the sum and reversing one of the resultants )
j 1 1 12 1 (l 1/ 2) ()2j () 2 j () 2 2 2[ (2j )] 2 . (B. 18 ) l 0 lj 2 j 482 jjj
Again, to first order, i n the case that = 2 , a very careful evaluation of limits yields j (l j 1) 2 1 2 1 (1j ) ( j 1)2 ( j ) 2 , (B. 19 ) l 1 l12 2 j which, by virtue of ( 23 ) , can also be written
j (l 1) 2 2 1 (1j )(22) jj ( 1)2 ( j 1) . ( B. 19 a ) l 0 l j 1 12
By incrementing the argument of the digamma function on the left - hand side and identifying most of the resulting sums, an extension of ( B. 19 ) is found j()ln n j 2 ( jnl ) 1 ( (1j ) ( n ))2 ( j 1) 2 l n1 l 2 l1 l 0 (B. 20 ) 2 1 ()n () n if nj 2. 2 12 In the case that = 1 , ( B. 20 ) reduces to the known result j 2 j (l 1) (1) 1 1 t 1( (1 j ) )2 1 ( j 1) log(1 t ) dt , (B. 21 ) 2 2 0 l 1 l12 1 t t he integral arising from Hansen [ 12 , Eq. (55.2.1)]. For the case = 0 , see ( B. 66 ). Note that , if the upper limit becomes negative in ( B. 20 ),
n j 2 ()jnl 1 () jnl if n j l0nl1 lnj 1 ln 1 0 ifn j 1. (B. 22 )
Considering the second order ter ms of the case = / 2 in ( B. 16 ) l eads to
25 Oct. 1 5 , 2017.
j(j l 1) j ( l ) 2 (1 j )( ( j 1)) 2 (3) (2) (1 j ). (B. 23 ) l1l l 1 l Or, if = − 1 in ( B. 16 ) , the second order terms lead to
j2(l 1) j 2 ( l j 1) 2 5 ()j () j2 (3) () j ( B. 23 a) 3 3 l0(1l j ) l 0 ( l 1) 3 12
For other variations reverse the sums, or apply ( B. 44 ) below.
From the literature [ 37 , Eq. (1.6) , corrected for the wrong sign in the first term of the right - hand side] by considering even and odd terms , and after applying ( B. 67 – see below) we obtain
n 1 n 1 1 2 2 (l ) n (l ) 2 2 ( ) (ln(2) (n )11 ( n )) ( 1 n 1 ) 11 ( ) ( 1 22 2 22 ) l1l n k 1 l 2 12
11 112 (B. 24 ) (22 )(2 (n ) ln(2) ( n )) ln(2)(2 ( 22 n ) n ) ( nn ) ( 1) 2 1 1 2 2() 2 n n (), n 6 and , from [ 37 , Eq.(1. 8 ) ] using the same method, we find
n11 n 1 1 ()l2 () l 2 1 12 1 1 2( 2 ) (3 (n 1) 4 (2 n 1) 2ln(2) n ) ( 2 ) l1l n l 0 l 1 21 2 2 6ln (2) (2 (nn 1) 8 (2 1))ln(2) 2 ( nnn 1) 4 ( 1) 2 (2 1) (B. 25 ) 2 1 1 4 (2n 1)2 ( 2 n ) . 4 Operating on ( 28 ) with , and evaluating the limit a b 1 leads to a k()(blbl 1) k () bl 2 () bl {2 } l0()bl l 0 bl () bl (B. 26 ) (bk )2 ( bk ) 1(()())bkb3 3 1 (() (2) bbk (2) ()). 3(bk ) ( bk ) 2 6 Use (B. 9 ) with n= 0 , s= 2 to obtain a variant of the above – see ( B. 8 ). For the case a b see ( B. 31 ).
As has been noted, operating on well - known hypergeometric sums is a technique extensively utilized in the literature [e.g. 17 , 8 ] . Typically, such calculations are based on variants of Gauss’ evaluation of any 2 F 1 (1) and the well - known Dixon/Whipp le/Watson evaluation (s) of a particular set of 3 F 2 (1) (e.g. [ 29 ], [ 33 ] ,[ 38 ] ) . Here, it is noted that other known 3 F 2 (1) evaluations lend themselves well to this technique . C onsider the tabulated E ntry 36 ( see Milgram, [ 23 ] ) , specifically 3 F 2 (1,1 - n, a;b,c;1) satisfying
26 Oct. 1 5 , 2017.
n1(1)(l al ) (1 aca )()sin() c n 1 (2 lcnbl )( 1) l0(nlblcl )( )( ) (1)(1) b nb l 0 (1)( l ac 2 nl ) (B. 27 )
Apply the ordered operator lim lim to both sides of ( B. 27 ) and, after considerable c1 na b 1 c a calculation involving the sums listed herein , eventually arrive at
n1(a l ) n 1 ( l 1) 1 ( (lnl 1) ( )) ( a ) 3 2 l0()al l 0 ()6 al n 1 (l 1) 1 3 (()(aan )) ( an )(( an ) ())( aan ) l 0 (a l ) 3 (B. 28 ) 1 1 1 ()(aan )2 (() a ( ana ))() ( (2) ( an ) (2) ()) a 2 2 3
In the case that = ( 1 − ) / 2 , (B. 28 ) reduces to an equivalent, but simpler form of ( 47 ) when = 2 , that being
2 j (l ) 1 1 ( j ) 2 /4. (B. 29 ) 1 2 2 l 1 2 j l If = 1 − , (B. 28 ) reduces to ( 18). Alternatively, apply the ordered operator lim lim c1 nb a a to ( B. 27 ) and again , after a long series of simplifications involving many of the sums listed here, as well as the following result, ( equivalent to [ 32 , Section 2, Theorem 1 with l 1 ] , and generalizing [ 31 , Eq. ( A.7 ) ] by the replacement a k ) n 1 (al ) ( an ) ()(1 a aaan )()( 1)( an ), (B. 30 ) l 0 find n 1 (al )2 ( an 1)( an ) 2 (12( an ))( an ) l 0 (B. 31 ) (1aa )()2 (2 a 1)() an 2.
For related results of the above form see Wei [ 31 , Eq. (A.4) ] and Wei et. al. [ 32 ] , and for those forms involving alternating series with binomial coefficients, but lacking the independent parameter “ a ” , see Choi [ 4 ] .
Similarly, consider the evaluated Entry 25 of Milgram, [ 23 ], that is 3 F 2 (a,1,b ; m+1,c;1) ( originally ( 1 2 ) ) . Operate on both sides with lim lim to eventually obtain c1 aa b 1 m b
27 Oct. 1 5 , 2017.
m 1 (al ) (1 l ) ( a 1) ( ( (a m 1) ( a 1)) l 0 (mll )( 1) ( m 1) 1 1 (B. 32 ) ( (am 1) ( a 1))2 ( ( am 1) ( a 1)) ). 2 2
See also ( B. 27 ) and ( B. 50 ) as well as Hansen [ 12 , section 55.4 ] . Note that the operator lim a m a applied to (B. 32 ) yields ( B. 58 ) below.
A related sum can be found based upon a variation of ( 16 ) , as well as ( 18 ), that being
m 1 (al ) (1 l ) (2 l ) ( a 1) 1 g 3( 2 ( (0) (0) (1) ) l 0 (mll )( 1) ( m 1) 6 2 m1(1l ) m 1 ( al ) (1 l ) ( (a 1)(0) ) l0aml1 l 0 al (B. 33 )
12 1 (2)( (m 1) (1) ) (0) ) 3 2 w here (also see ( 9 ) ) ()n () n (n ) (am 1) (1 ag ) and ( m 1) ( a 1). A related result follows:
((1))(1)mmallm1 ( )(1) m 1 (1) l ( (a 1) ) (a 1)l0 ( mll )( 1) l 0 aml 1 m1(aml 1) (1 l ) m 1 (1 l ) 2 l0aml1 l 0 ( aml 1) (B. 34 ) (m 1)m 1 ( al )(1 l )(2 l ) (a 1)l 0 ( mll )( 1)
Eliminating terms between (B. 33 ) and (B. 34 ) is equivalent to (B. 32 ). Similar results may be obtained from the well - known, simple hypergeometric identit ies (see also (B. 61 ) )
n 1 ()la ( na ) ( ( a ) ( n a )) l 0 (lnl ) ( 1) (1 n ) (B. 35 ) and l n 1 ( 1) a . l 0 (nlla )( 1) ()(1 n a )(1 na ) (B. 36 )
A fter differentiating once with respect to a , the first yields
n 1 (la )( la ) ( na ) 2 (()( a na ) ( na ) (n a ) () a ) , l 0 (lnl ) ( 1) (1 n ) (B. 37 )
28 Oct. 1 5 , 2017.
and t he second yield s the known result l n 1 ( 1) (la 1) a (1 a )(1 na ) 1 n . l 0 (nl ) ( l a 1) (1na )2 ( n ) (1 a ) (B. 38 )
Comparing the zero th order terms in the limiting case → − 1 gives the known result
n1 n 1 (l 1) ( n l ) 12 1 2 1 2 ()n ()/ nn () n /12 . l0l1 l 0 n l 2 2 2 (B. 39 )
Additionally, the first order terms in this limiting case give
n2(nl 1)2 n 2 ( nl 1) ( n 1) 2 (n ) ( n 1) l01nl l 0 1 nl (n 1)
12 1 2 (3) (B. 40 ) ((n 1)3(2) ( n 1)) 3 . 3 633
With respect to the former case, see ( 23) and (B. 9 ) ; otherwise , see (B. 44 ) , ( 52 ) and ( B. 59 ); regarding either case, reverse the order as illustrated throughout .
A lternatively , consider the aforementioned Entry 25 under the ordered substitution = 1 , → 1 − , → 1 − . Applying the ordered operator lim yields c a c a n 1 ( 1)l (a l )2 ( a 1) (a 1) 2 ((a 1) ( an 1 )2 2()) n l 0 (nll )(2) na ( 1 n ) (n 1) (B. 41 )
Or , d ifferentiate twice with respect to the parameter “ a ” , followed by → 1 to obtain n1(1)(l2 cl ) n 1 (1)( l cl ) l0(1)(l nl ) l 0 (1)( lnl )
(B. 42 ) (1 c ) 1 3 (2) 2 2 2 (( ) [ ] ) (n 1) 3 CC CC6 CC w here
C ()c ( c n ). In the limit → − , (B. 42 ) reduces to a variant of ( B. 58 ) , but also see ( B. 47 ) .
U sing Entry 36 of Milgram, [ 23 ], that is 3 F 2 (1,1 - m, b ; c, 1 - β ;1) , a pply the ordered operator lim lim to obtain c1 b 1 c b
29 Oct. 1 5 , 2017.
m1l 2 m 1 ( 1) (1 l ) (m 1 l ) 2 m ( )() m l0(mll ) ( 1) l 0 ( l 1) m 1 2 ((1 ) ())(()m m )(1 m ) 2 m (B. 43 )
21 ()1m 2 (2 )(1) (() m (1)) m m m 2 th where m 1 m . In the case m 0 , equating the coefficients of the zero and first order terms respectively gives
m 1 2 m 2 (1l ) (1 l ) 1 3 2 (m 1)( ( mmm 1)) ( 1) ( 1) ( m 1) l0lm l 0 ( l 1) m 1 3 (B. 44 ) a nd
m 1 (lml 1)2 ( ) m ( l 1) 1 m (2) ( l 1) 4 (m 1) l0ml l 0 l 1 2 l 0 l 1 120 2 22 4 (3) (mm 1)(3 ( 1)5 /6 )/2 ( m 1)/2 ( m 1)/12 (m 1)(( 1/ ( mm 1) ) ( 1)(2) ( m 1)/ 2 2 (3) ) (B. 45 ) (m 1)/22 2 ( m 1)/6( 1/( m 1)) (2) (m 1)/ 2 .
By differentiating ( B. 43 ) once with respect to , a lengthy , but general, result for the sum
m (l 1)2 ( l 1) ( B. 43 a) l 0 (m l 1) ( l 1)
can be found. Because of its complexity, th at full result (whose simplification, if any exist s, eludes me) is presented in the form of computerized output – see Appendix D . In the case m 1 , we find m m (l 1) 1 (1)(lmlm2 1)( 1) ( m 1/2( mm 1)( 1)) 2 l0 l 0 (l 1) 3 (m 1)( 2 ( mmmm 1)) ( 1) ) ( 1) ( 1)3 (3 mm 1) ( 1) 2 ( B. 46 ) (1 6m ) ( m 1) 6 m , and for the case m see ( B. 45 ).
Obvious v ariations of the above can be obtained by reversing the sum on the left - hand side or applying any combination of ( B. 8 ) , (B. 10 ) and/or B. 58 ) ; note also the parallels between ( B. 44 ) and ( B. 16 ) taking ( 21 ) into account or (B. 28 ) when a=1 . Also, comparing ( B. 44 ) with (B. 42 ) in the case that → − leads to
30 Oct. 1 5 , 2017.
n2(1l )2 n 2 (1 l ) 2 1 ()n2 (2() n )( 2 6()) n l0l n 1 l 0 ( l 1) 6
3 2 1 4 (B. 47 ) (3) (2) (n ) ( n ) 3 . 33 3 3 4 In the limit n the d ifference of the sums in ( B. 47 ) diverges as ln3 (n ) . Further, ( 21 ) with 3 b= 1 applied to ( B. 47 ) leads to
n2(1l )2 n 2 (1 l ) 1 ()n2 (2() n )( 2 6()) n 2 l0l n 1 l 0 ( l 1) 6 1 (B. 48 ) (3) (2) (n ) ( n ) 3 , n 2 2 Compare with ( B. 40 ). Further variations can be identified by splitting any of the above the sums according to whether n is even or odd. Specifically , in the case of ( B. 47 ) , for n> 0 and n 2 n we have
22nn(1l )22 22 (1 l ) n 1 (1 l ) 2 n 2 ( ln 1) 2 2n ( 2 2 ) , (B. 49 ) ll00ln2 1 ( l 1) l 0 ( lnl 2 1)( 1) l 0 ( ln 1)
and, in the case that n2 n 1 we find
2121nn(1l )22 (1 l ) n (1 l ) 2 n 2 ( ln 2) 2 (2n 1) ( 2 2 ) . (B. 50 ) ll00ln2 (1) l l 0 (2)(1) lnl l 0 (3/2)(1/2) l n
Another method that can be used to obtain useful sums is to make use of known hypergeometric (Thomae) transformations of any 3 F 2 (1) . Consider Ent ry 25 - see (B. 42 ) - and Entry 2 6 – 3 F 2 (a, n+1,b ; n+2,c;1) - ( both listed in [ 23 ] ) which can be transformed into one an other by careful use of limits. Let c, b , nn 1 in both entries, and evaluate the limit a n 1 in Entry 25 and the limit a 1 in Entry 26 . The left - hand sides of both entries are now equal, so equating the right - had sides of each and replacing gives a useful identity with two free parameters : n ( 1)l (l 1) (1 l ) l 0 (l 1) ( nl 1) ( ln ) n 1 ( 1) (2l ) ( ()(n 1)(2 )sin( )( nlnl 1)l 0 ( )(2 ) (B. 51 ) ( 1)n (2n )( ( n 1) ( 1) ( n 1) ) ).
In the case 0 , n 1 , (B. 51 ) reduces to
31 Oct. 1 5 , 2017.
n (1)(1)l l () ( n ) . l 0 (nl 1)( l 1)( n )()(1) n (B. 52 )
Differentiate ( B. 52 ) once with respect to β to obtain
n ( 1)l (l 1) ( l 1) (()(n )()1/( )( n ) ) () ( n ) l 0 (nll 1)( 1) ( nn )()(1) (B. 53 ) a nd, in the limit n 1 , comparison of the three lowest order coefficients yields n (l 1) ( ( n 2) 1)( n 1), l 0 (B. 54 )
n (lnlnn 1) ( 1) ( 1)( ( 2)2 ( n 2) 2 ( n 2) 2 2 / 6) l 0 (B. 55 ) and n n (l 1) ( nl 1)2 ( l 1) ( nl 1) l0 l 0 (n 1) (3(n 2)3 9( n 2) 2 (2 n )189( (2 n ) 2 ) 3 (B. 56 ) 9 (2 n )3 (2) (2 n ) 6(3) 2 18).
( B. 54 ) is a well - known result [see Hansen 12 , Eq. ( 55.6.1 ) ], while ( B. 55 ) has the interesting property that the left - hand side is invariant under reversal , so, from ( 67 a) we find nn n ll( 1) ( nl 1) ( l 1) ( nl 1). ( B. 55 a) l02 l 0
By reversing either or both series, ( B. 56 ) can be identified with several different combinations using the identities n n (l 1) ( nl 1)2 ( l 1) 2 ( nl 1) l1 l 1 and ( B. 57 ) n n (l 1) ( nl 1) ( l 1) ( nl 1). l1 l 1
In a similar vein, consider the limit n, n 0 in ( B. 53 ) and for the latter two cases employed
previously , (the zero th order case reduces to ( B. 16 )) obtain
n 1 2 (1)()1lnl 3 (1) n 2 1 (2) (n 1) (3 ( n 1) ) ( n 1) (3), l 0 ()2n l 2 22 (B. 58 )
32 Oct. 1 5 , 2017.
a nd
n1(1)(l nl ) n 1 (1)( l nl )2 1 (n 1) 4 l0()nl l 0 () nl 3 2 2 2 2 (nn 1) (2 ( 1) ) ( n 1) ( n 1) 6 6 (B. 59 ) (n 1) 2 (3) ( n 1) 4 (4(2) (n 1) 8 (3) 3 ) . 3 2 3 45
Again, by reversing each of the sums using the identities
n1(1)(l nl )2 n 1 (1)( l 2 nl ) l0n l l 0 l 1 and ( B. 60 ) n1(1)(l nl ) n 1 (1)( l nl ) l0n l l 0 l 1 a variety of sums can be identified. See also the note preceding ( B. 47 ).
B ased on the hypergeometric identity ( 16 ) and its reversed form, that is
n1()la () na n 1 (2 nal ) (a 1 n )( ( n 1) ) 2 ( ) l0(lln 1)( ) ( n 1) l 0 ( l 1) (B. 61 ) d ifferentiating once with respect to a gives
n1()()()lalana n 1 (2 nal ) n 1 (2 nal ) (n a ) 2 ( l0(llnn 1)( ) ( 1) l 0 ( l 1) l 0 ( l 1) (B. 62 ) ((1)(a nna ) (1) a n )( (1) n ) ) and in the limit a 1 , we find
n1(1)l n 1 () nl n 1 () nl (n 1) ( (1 nnn )2 (1 ))( ( 1) ). 2 l0(ln ) l 0 ( l 1) l 0 ( l 1) (B. 63 )
In a sense, this generalizes ( B. 3) .
F rom the literature , Speiss [ 30 , L emma 18] ( presented as a sum of harmonic numbers , rewritten here ) provides the following transformation
33 Oct. 1 5 , 2017.
k(l 1) m ( l ) ( k 1) 2 1 2 ( (k 1) ( m 1)) l0lm1 l 1 lkm 11 lm 6 (B. 64 ) 1 2 ( (k 1) ( m 1))( ( k 1) ( m 1) 2 ( km 1)), w hich, in conjunction with ( 18) using b 1 and c m 1 , produces the variation
m(l ) k ( l m 1) 1 ()mkm ( 1)(() m 1 ) (() mk ()) 2 l k l 1k 1 2 l1 l 0 (B. 65 ) 1 (m ) ( k 1) 2 (()())k m . 2k 6 In the case that k 0 we obtain ( also see (B. 13 ) and ( B. 20 ) ) m 2 ()1l2 2 () m (()m () m ) . l 1 l2 6 m (B. 66 ) And, by considering its even and odd terms, we find the related alternating sum m 1 m 2 m (l ) (l ) ( 1) m l 1 2 1 m( 1) 1 m 1 (1) [()m () m2 ()]()[ m 2 ()] m l 2l 1 2 2 2 l1 l 1 2 (B. 67 ) 2 2 m ( 1) 1 m1 22 1 m 2 2 [ ( )(ln(2 ) m ) ln ( 2 )m ln(2)] 4 ( ) . 2 2 24 4 Finally, a s discussed following ( 69 ) we find
m1(l 1) m 1 ( l 1) 1 2 1 (2) 2 2 (6 (m 1))( ( m 1))2 ( m 1) (3). (B. 68 ) l0(l m ) l 0 ( l 1)
See also ( B. 6 ) , ( B. 10 ) , ( B. 48 ) and ( B. 63 ).
Appendix C : Infinite Sums
The following useful and related infinite sums, based on the methods of this paper, and needed (or obtained) elsewhere [ 19 ], are presented * with minimal proof.
1 1 1 2 2 (C. 1 ) ( ) ( )( ) 2 7 ( 3 ) 1 3 n 0 n n 0 n n 1 3 1 1 2 2 ( 2 ) 2 ( 2 ) 1 1 1 ( l 3 ) ( l 1 ) 2 ( ) ( )( ) 2 2 1 3 1 (C. 2 ) n 1 0 n 1 2 n 2 0 n ( 2 ) 1 n ( 2 ) 2 l 0 l 2 3
* n ( 2 ) n 1 n 2 ; n ( 3 ) n ( 2 ) n 3
34 Oct. 1 5 , 2017.
1 1 1 1 n ( 3 l ) ( 1 ) 2 (C. 3 ) ( ) ( )( ) 2 2 1 5 3 ( ) n 1 0 n 1 2 n 2 0 n ( 2 ) 1 n n ( 2 ) 2 ( n 2 ) l 0 ( l 1 ) 6
( l 1 ) 2 ( l 1 ) 2 ( l 1 ) ( l 1 ) 2 2 (C. 4 ) 2 2 ( 1 ) 1 2 l 0 l 2 3 1 1 [ ( 3 n ) ( 5 n ) ] 1 ( 2 ) 2 ( 2 ) n 0 ( n ) n 0 ( n n 1 ) 1 1 2 2 1 2 1 1 1 1 2 (C. 5 ) 1 ( ) ( ) ( )( ) 4 log( 2 ) 3 ( 3 ) 2 1 5 2 n 1 0 n 1 2 n 2 0 n ( 2 ) 1 n 3 0 n ( 3 ) 2 n ( 3 ) 3 3
Evaluate p q in ( 37 ) : (l m 1) 2 ()(q qm ) ()( q qm ) l 0 (l q ) m 1 (l 1) 1 (2) ()q () q () q 2 2 (C. 6 ) l 0 (q l 1) then if q m :
( l m 1 ) 2 m 2 ( l 1 ) ( m )( ( m )) ( 2 ) ( m ) ( m ) 2 2 l 0 ( l m ) 6 l 0 ( l m 1 ) (C. 7 )
( a l n ) ( a l q n ) [ ] l 0 ( a l ) ( a l q ) (C. 8 ) n 2 ( a l 1 ) { 1 ( ( a ) ( a ) 2 ) ( a )[ ( n ) ( 1 )] } { a a q } 2 l 0 ( l 1 ) Generalizations of ( 56 ) : (lkj1 ) ( lk 1) [2 ] 1 l 0 l k 2 12 k 1 (l 1) ()12 ()(k 11 ) 11 () k 2 2 22j 0 22 1 2 12 l 0 (l 2 ) (C. 9 ) j 2 (k l3 )() k 1 2 2 k 0 l 0 (l 1)
[ (lN 1) ( lj 1)] l 0 (l y ) N j 1 (j 1 k ) (C. 10 ) ()(yN( 1 y ) (1 j yNj ), ) k 0 (j 1 k y )
[ ( l x N 1 ) ( l x 1 )] N 1 ( x k 1 ) ( y ) (C. 11 ) l 0 ( l y ) k 0 ( x k 1 y )
35 Oct. 1 5 , 2017.
x , y 0 , 1 ,
[ (lxN 1) ( lx 1)]N 1 ( xk 1) ( x ) l0(l x ) k 0 ( k 1) (C. 12 )
x 1 , 2 , From ( 62 ) and [ 13 , Eq. 9) * ]
( 1 l ) 2 7 ( 3 ) 2 log( 2 ) . 1 2 (C. 13 ) l 0 ( 2 l ) Evaluate the limit k in (B. 9 ) to obtain a variation of ( 66 ) (n ) ()b l( 1) n 1 1 n ! ()n()b 2 (21) n (),0 b n . n 1 2 { } l 0 ()bl n ! (21)! n (C. 14 )
See the note following ( 60 ) for the (divergent) case n=0.
In ( B. 12 ) w hen k we find (1 l ) 7 4 2 2 2 5 . (C. 15 ) l 0 (1l ) (2 l ) 180 6 F rom [ 19 , Eqs. (5.1.7), (5.1.10) and (2.10)]] we have
t ( 1 ) l 2 2 8 log( 2 ) ( l 1 )( l 3 )( l 2 ) 3 l 0 2 (C. 16 ) and t ( 2 ) l 2 2 16 log( 2 ) 2 ( 3 ) 3 5 3 3 (C. 17 ) l 0 ( l 2 )( l 2 )( l 3 ) where [ 20 ] 3 1 1 tl(1) ( l 2 ) ( 2 ) H l 1 ( 2 ) l 3 1 (k 2 ) () 2 (C. 18 ) t l (2) . k 0 (k 1) From the Thomae transformations [ 18 , section 3.13.3] as employed in ( 51 ) , variations of ( 52 ) are: ( l 1 ) ( q 1 l ) 2 ( q ) 2 (C. 19 ) l 0 ( q l ) ( l 1 ) 6
( l 1 ) * which reads: 2 ( 7 ( 3 ) 2 ) / 2 - s ee ( C. 28 ). 1 2 l 0 ( l 2 )
36 Oct. 1 5 , 2017.
( l 1 ) 1 , 1 , 1 , 1 1 ( 2 ) ( q ) ( q ) ( q ) 2 ( q ) 4 F 3 ( | 1 ) / q l 0 ( q l ) 2 , 2 , q 1 (C. 20 ) 2 1 ( 2 q l ) ( l 2 ) 2 ( 3 ) ( ( q )) 6 ( 1 q ) l 0 ( 2 l )( 1 l )
( l 1 ) ( q l ) ( q ) 1 ( 2 ) ( q ) (C. 21 ) 2 2 l 0 ( q l ) ( q l )
( l 1 ) ( l q ) (C. 22 ) ( q ) ( q ) ( q ) l 0 ( q l )
( l 1 ) ( 2 q l ) 2 2 ( q ) cot( q ) ( q ) 1 ( 2 ) ( q ) 2 2 (C. 23 ) l 0 ( q l ) ( l 1 ) 6 6 1 Therefore, comparing (C. 19 ) and ( C. 23 ) at q 2 gives (1 l n ) 2 1 2 ()n 1 () n 11 (2) () n 1 21 2 2 22 2 l 0 (1l ) 6 ( n 2 ) (C. 24 ) n 1 (1 l ) 1 2 7 (3) 2 , 2 2 1 2 l 0 (2 l ) ( 1 l ) 7 ( 3 ) , 1 2 (C. 25 ) l 0 ( 2 l ) a nd
(1 l ) 1 ln(2) 2 2 ( )8ln(2) 7 (3) . 2 2 (C. 26 ) l 0 (1 l ) 3 6 3
Equation (C. 24 ) generalizes [ 9 , Eq . (15)] . A nother more general form of the above appears in [ 13 , Eq. (8a)] , that is (1 l ) 2 ()(2)1n 1 (2 n 1)(22n 1 1) 2 n 2 2 l 1 l n 1 (C. 27 ) (22n 1 2 l 1)(2)(2l n 1 2), l l 1 along with its related form (l 1 ) 2 (2)(2n2n 1)1 (2 n 1)(2 2 n 1 1) (l 1 ) 2 n 2 l 1 2 n 1 (C. 28 ) (22 l 1) (2)l (2 n 1 2). l l 1
Applying the limit k to ( 21 ) gives
37 Oct. 1 5 , 2017.
(lb ) ( cl ) 2 2 ()()(()bc b ())/( cbc ) ()/( bbc ). (C. 29 ) l 0 (cl ) ( lb )
For the limit b c see ( 62 ) . After o perating on ( C. 29 ) with the operator d we obtain dc (lb ) ( cl ) ()(bc ()1/( bc ))2 2(() c ())/( bbc ) 3 2 2 (C. 30 ) l 0 (cl ) ( lb ) ()/(c b c ) 2 and, in the limit c b (l b ) 1 ()b2 1 (3) () b (C. 31 ) 2 2 12 l 0 (b l ) From [ 19 ] and res olving * [ 12 , Eq. (55.6.4)] find:
x2l 1 [ (1 l ) (1)] 2 log(21x ) 2Li ( 1 x ) 1 log( 2 11 xx ) log( )log4 1 622 2 2 1x 1 x l 0 (l 2 ) (C. 32 ) d Operate on (C. 32 ) with to obtain d x ln(1 x ) xl ( (1 l ) (1)) (C. 33 ) l 0 (x 1) reproducing a known result [ 12 , Eq. (55.3.1)] ( x 1 ) .
From [ 21 , Eqs. (9), (10) and (12)] we have:
1 11 1 2 (qlql2 )( 22 )(( q 2 )()) q 1 ( (2qq ) (22 ))( (1 2 qq ) (2 )) (C. 34 ) l 0 (1l 2 q ) 4
equivalent to (C. 35 ) ()(ql ql 1 ) cot( q ) 2 ( (q1 ) ( q ))2 4 cot(2 qqq )( (2 ) (2 1 )) 12 2 1 l 0 (1lqlq )(2 ) ( q 2 ) and ( 1)l (q l ) ()(()q q ( q 1 )) (( 1 q ) ()) q 2 2 2 2 . (C. 36 ) l 0 (l 2 q 1) 2 8 1 The result ( C. 34 ) generalizes [ 9 , Eq(24)], which corresponds to the case q=0 . If q 2 (equivalent to [ 12 , Eq. ( 55.6. 2) when x i ]) then
( 1)l (1 l ) 2 2 ln(2)( 2ln(2)) 2ln(2), ( C. 37 ) l 0 (l 1) 8 which, by grouping terms in pairs can equivalently be written as
* notice that [ 12 ,Eq. (55.6.7)] is incorrect – omit the term “ 2x ”;
38 Oct. 1 5 , 2017.
(1 2l ) 2 5 1 2 ln(2)( 2ln(2)) 3ln(2). ( C. 38 ) 4 1 l 0 (l 1)(2 l ) 8 6
Applying the limit k to ( 28 ) with a b 2 gives
()bl ()() bb () b (b l ) 2 , ( C. 39 ) l 0 (al ) ( ab 1) ( a 1) ( aab 1)( 1) yielding the convergent series (b l ) 1 ( (b ) 1) ( C. 40 ) l 0 (blbl )( 1) b when a b 2 , generalizing [ 17 , corollary 2.3] . To ( C. 39 ) apply the ordered operator s lim and lim independently, then add the two results to eventually obtain a b 2 b a b 2 a
(b l ) 1 ( ()1)b () b ( C. 41 ) l 0 (blbl )( 1) b and 1 1 1 ()bl(2 2 ) (()/1)(1). bbb ( C. 42 ) l 0 (1)b l ( bl ) b
Useful sums deduced from among all the previously compiled entries follow:
(lj ( 1)/2) ( lj /2) 2 ( ) (/2)j2 (/2)((1)/2) j j 2 l 1 lj( 1)/2 lj /2 ( C. 43 ) ((j 1)/2) 4 ( j /2)/ j
(l j 1 ) ()(1)l j j 22 (2 ) ()(j j 1 ) ljlj11 j 12 jj 1 1 l 1 2 2 2 ( C. 44 ) 2 j 2 (l 1) ( 2ln(2))2 2 1 6 l 0 l 2 1 (l j ) (l j ) 1 2 (2 ) ( (jjj ) ( )) 2 (1 ) (2 2ln(2)) ( j ) l l 1 2 2 l 1 2 ( C. 45 ) 2 j 2 (l 1 ) 2 ( 2 2ln(2))( 2 2ln(2)) / 2 2 12l 0 l 1 and, by solving a single step recursion equation
39 Oct. 1 5 , 2017.
(l j ) (l j 1 ) 1 2 2()j (1 j )2 ( j 1 ) 2ln(2) 2 (1 ) (2 2 ) l 1 l 2 l 2 1 (2ln(2) 2) (1 j ) 2 /2 (2 4)ln(2) ( C. 46 ) 1 2 j 2 12 j 2 (l 1) (2 )4 . 1 3 j212 l 0 l 2 ( 1)l (l n ) By considering the odd and even terms of the sum individually and applying the l 1 l( l n ) above, as well as simple results summarized in Appendix E , the following two results for odd and even values of n respectively can be found :
( 1)l (l 2 j 1) (2j 1) 2ln2 (2) ( ( jjj 1/ 2) 2 (2 1) 2)ln(2) ( 1/ 2) l 1 l( l 2 j 1) ( C. 47 ) 2 (2j 1) 1j 2 ( l 1/2) ()j ( 2 () j ( j 1/2))/4 2 24 (2j 1) 2l 0 l 1 and l ( 1) (l 2 j ) 2 1 1 j ln (2) (1 (j2 ) / 2 / 2)ln(2) ( j 2 ) / 2 l 1 l( l 2 j ) 1 2 2 2 ( C. 48 ) ( (j 2 ) () j /6/8 ) ()/21 j /2 1j 2 (l 1) (j1 ) ( j ) 2ln(2) / (8 j ) . (2 ) 3 4 l 0 l 2
From a n unpublished source [ 11 ] , the following was contributed
1 1 1 2 l (12l ) ( 2 2 l ) 1 ln(2) ( 1)2 2 (3) ( C. 49 ) l 1 l 6 and ()11l m 7 1 ( 1)l m 4 (3) 4Li (1 ) (3)ln(2) (ln 422 (2) ln (2)) 4 2 , ( C. 50 ) m1 l 1 lm()2404 l m 2 6 the first of which (see also ( C. 36 ) and ( C. 37 )) can be proven using the results contained previously herein by considering its even and odd terms . The inner sum (s) of the second is given by ( C. 47 ) and ( C. 48 ) ; although ( C. 50 ) was offered without proof, it is numerically verifiable , and, temporarily accepting its validity , a number of other interesting and useful results can be deduced. Adding the even and odd terms of the inner sum of ( C. 50 ) and utilizing many of the results found in Appendix E, eventually gives
1n (1 m ) 1 n (1 m ) 2 (3 (1 ) 7 ln(2) 3) (3) 12 1 2 2 2 n1(n2 ) m 1 ( 2 mn ) n 1 (1) m 1 (1) m ( C. 51 )
5 12222483 4 1 ((ln(2)62 ) ( ) 2ln(2) 3 ln (2)) 3 ln (2) 16Li 4 ( 2 ) 360
40 Oct. 1 5 , 2017.
or, after employing ( 23 ) 1 1n (m 1) (7 (1 ) 7 ln(2)) (3) (12 2 ) 1 2 n1(n2 )(1) n m 1 ( 2 m ) ( C. 52 )
418 1224 179 4 1 ((ln(2)3233 ) () ln(2) ln(2)) 3 ln(2) 8Li 4 (). 2 360 Subtract ( C. 51 ) from ( C. 52 ) to obtain 1n (m 1 ) (m 1) 2 (11 (1 ) 14ln(2) 3) (3) 2 (1 ) 2 2 n1(n 1) m 1 (1 m ) (2 m ) ( C. 53 ) 1121224 1 1 4 1 (223 ( ) ln(2) 3 ln (2)) 3 ln (2) 90 8Li 4 ( 2 ), A gain a pply ( 23 ) to the inner sum, and notice that all the sums so - created appear in Appendix E . A fter considerable computation an d simplification , it is found that ( C. 53 ) reduces to an identity, thereby verifying ( C. 50 ).
Alternatively, r e order ( C. 52 ) along a diagonal of the summation grid, interchange the summations , simplify using results from Appendix E and eventually obtain (l ) (2 l 1) 79 7 ( (1 ) ln(2) 1) (3) 1 ln4 (2) 4 2ln 2 (2) 5 1 4 2 12 l 1 (2 l ) 1440 ( C. 54 ) 1 125 1111 2 2 1 (4 (ln(2) 2 ) 2ln(2) 2 ) ( 22212 ) ( ln(2) ln (2)) 2Li 4 ( 2 ).
Appendix D (A Special Case)
The following is the computer output defining the result associated with the generalization of ( B. 43 ) after differentiating with respect to the variable β. In the following, the variable
B m and the various parameters Y n are defined following the main result. For the special case s m and m 1 , see ( B. 45 ) and ( B. 46 ) .
41 Oct. 1 5 , 2017.
Appendix E (Simple S pecial C ases)
The following are commonly needed, useful and interesting cases that can be derived from the more general entries listed above , or extracted from the literature .
(l 1/2) 7 * (3) 2 /2 (E. 1 ) 2 l 0 (l 1/2) 2
(l 1) 2 (E. 2 ) 2 ( 2ln(2)) 7 (3) (l 1/ 2) 2 l 0 (l 1) 1 2 (3) (E. 3 ) 2 l 0 (l 1) 6
4 (l 1) (E. 4 ) † 3 (3) l 0 (l 1) 360
(l 1) 4 (E. 5 ) 7(2ln(2) ) (3) (l 1 )3 8 l 0 2 (l 3 ) (E. 6 ) ‡ 2 1 2 ( / 2 ln(2)) 7 (3) (l 1) 2 3 2 l 0 (l 1) 7 § (3) (E. 7 ) 1 l 0 (l 2 ) 2
(l 1) 4 4 22 28ln(2) (3) 151 4ln (2) 4 ln (2) 32Li (1 ) 1 2 360 3 3 4 2 (E. 8 ) l 0 (l 2 )
(l 1 ) 5 4 (E. 9 ) ** 2 (l 1 )2 24 l 0 2 (l 1)2 11 4 22 (E. 10 ) †† 2 2 (3) (l 1) 360 6 l 0 1 ‡‡ (l ) 2 7 (E. 11 ) 2 ln(2) (3) 1 l 0 (l 2 ) 2
* Repeating footnote to ( C. 13 ). † See ( 72 ). ‡ Repeating ( C. 32). See also [ 9 , Eq(15)] . § Repeating ( C. 25 ). ** Special case of ( C. 31 ) †† See ( 72 ) and [ 9 , Eq.(9)]. ‡‡ Repeating ( C. 13 ).
42 Oct. 1 5 , 2017.
1 * (l 1) (l ) 2 2 2 (E. 12 ) 2 1 ( 2ln(2)) /3 /2 2 4ln(2) (1 ) 2 l 1 l l 2
(l 1 ) (l 1) † 2 2 (1 ln(2)) 2 /6 4ln 2 (2) (E. 13 ) (1 ) l 1 l l 2
(l 1 ) (l 1) 8 2 4 (E. 14 ) ‡ 2 32ln(2) (l 1)2 ( l 1 ) 2 3 12 l 0 2 (l 1) 1 2 4 ln(2) 4ln 2 (2) 1 3 (E. 15 ) l 0 (l 1)( l 2 )
(l 1 ) § 2 1 24( 2)ln(2) 8ln 2 (2) 4 (E. 16 ) 1 3 l 1 l( l 2 ) (l 1) l 1 2 (1) 2 ln(2) ln(2). (E. 17 ) l 0 (l 1) ( 1)l (1 l ) 2 ** 2 ln(2)( 2ln(2)) . (E. 18 ) l 1 l 8
(1 l ) 9 ( 1)l 2 1 2 (3) (E. 19 ) 2 12 l 1 l 8 (1 l ) ( 1) l 2 17 4 2 360 (E. 20 ) l 1 l
The following are useful results extracted from [ 2 ] , [ 7 ], [ 9 ], [ 10 ] , [ 25 ] and /or [ 37 ] . From the general result [ 25 , Eq. (3.11)] we have (l 1) ( 1)l 1 2 5 (3). 2 12 8 (E. 21 ) l 1 l Q uoting [ 9 , Eq . (2 1 )]
1 2 (2 l ) 7 122 1 22 3 4 (3) ln(2)(2 ) ( 4 4ln (2)) 4ln (2), (E. 22 ) l 1 l and from [ 25 , Eq. (4.12)] or [ 7 , Eq. (38)]
(l 1) 2 (3), (E. 23 a ) l 0 l 1 (l 1) l 1 1 2 ( E. 23 b) (1) 4 (3) 4 ln(2), l 0 l 1 along with , as a consequence of [ 10 , Theorem 3.1] or (C. 14 ) ,
* Special case of ( C. 43 ) † Special case of ( C. 44 ) . ‡ See ( E. 8 ) . E. 8 § From [ 37 ], second unnumbered equation following (1.9) , corrected for an incorrect exponent minus sign ** See ( C. 37 )
43 Oct. 1 5 , 2017.
(l 1) 1 4 . (E. 24 ) 2 120 l 1 l
From corrected [ 9 , Eq . (22)] , ( apply a factor of two to the right - hand side; also see (E. 6 ) ) 1 2 (2 l ) 41 2 2 1 2 /8 ()2 /6 7 ()(3). 2 (E. 25 ) l 1 l 7 Similarly, from the sixth (corrected) Example (3.6) of [ 37 ] – (remove the coefficient 2 ; it should be unity) we obtain 2 (1 l ) 61 422(2 ln (2) 2 ln(2) 14 24 ) ln (2) 14 (3) 4 2 32Li ( 1 ). (E. 26 ) 1 2 3 234 2 l 1 (l 2 ) 360
From corrected [ 9 , Eq . ( 19 )] , ( apply a minus sign to the right - hand side ) or [ 25 , Eq. (3.11) with k=2 ] 1 l (2 l ) 1 2 ( 1)2 7 (3)/2 2 G (2 ) /12. (E. 27 ) l 1 l 2 From corrected [ 9 , Eq . ( 12 )] , (replace ½ ζ(3) with ¼ ζ(3) ; also in [ 9 , Eq . ( 11 )] replace 12 with 2 ln(2) 12 ) , and ( E. 17 ) 2 l (1 l ) 13 22 2 1 ( 1) 3 ln (2) ln (2) ( /12 )ln(2) 4 (3). (E. 28 ) l 0 l 1
From corrected [ 9 , Eq . ( 8 )] , ( to both Eq. (8) and previous unnumbered equation , replace 1 2 1 2 2 12 ln(2) with 12 ln (2) ) or [ 7 , Eq.(30)] we have
4 l (l ) 17 1 2 21 4 3 ( 1)3 2Li 4 (2 ) 4 ln(2) (3) 12 ln (2) 12 ln (2) 4 (3), (E. 29 ) l 1 l 48 and from corrected [ 9 , Eq . ( 13 )] , (contains the same error as [ 9 , Eq . ( 8 )] – see above)
2 l (l ) 11 4 42 22 ( 1) 2Li (1 )1 ln (2) 1 (7ln(2) ) (3) ( ln (2) ). 2 4 212 4 12 (E. 30 ) l 1 l 480 By considering the even and odd terms of ( E. 29 ) we obtain
(l 1 ) 2 (3)(14ln(2) ())1 222 ln(2) 2 ln(2) 453 4 16Li ()1 (E. 31 ) 3 2 3 3 360 4 2 l 1 l and from the second unnumbered equation preceding [ 37 , Eq. (1.5)] we have (see ( 11 ) )
(1 l ) 1 2 2Li ()1 (7 ln(2))(3) 41 22 ln(2) 1 ln(2). 4 (E. 32 ) 3 4 2 4 12 12 l 1 l 120 By considering its even and odd terms, we also find
44 Oct. 1 5 , 2017.
1 l (l ) 4 2 2 4 (1)2 (3 7 ln(2))(3) 11 1 ln(2) 1 ln(2)2 Li ().1 (E. 33 ) 3 4 4 1440 12 12 4 2 l 1 l E xtracted from [ 10 , Section 8.2] , (l ) (1)l (1 /2 ln(2)) 2( G ) 4 2ln(2) (E. 34 ) 1 l 0 l 2 and 3 l (l 1) (3) 4 3 ( 1) 1 (1 ) 7 (3) ln(2) . (E. 35 ) 1 3 324 4 22 4 l 0 (l 2 ) w here the trigamma function arises from [ 14 ] and the imaginary part of Li 4 (i). Also see ( C. 27 ) and C. 28 ).
By rewriting, and disentangling the remain ing (corrected – also see ( E. 26 )) cases presented in 41 [ 37 , Example (3.6)] (in the fifth case of this example collection , change the coefficient 16 to 61 16 ) we find , after considerable effort, the follow ing relations (1 l ) 2 1 2 5 (3), (E. 36 ) 2 6 8 l 1 l (1 l ) 2 2 (3)(ln(2) 5 (1 ))1 ln4 (2)1 22 ln (2) 2 2 ln(2) ( 1 ) 2 4 262 3 2 l 1 l (E. 37 ) 121 2 1 4 1 6 ()2 120 4(),Li 4 2
(1 l ) () l 2 1(3)(3 (1 ) ln(2)) 1 222 ( 1 ) 2 ln(2) ( 1 ) 17 22 ln (2) 2 82 6 2 3 2 24 l 1 l (E. 38 ) 1 41 4 1 24 ln (2) 32 Li 4 (2 ),
1 2 ( l ) 2 2 (1 )2 ( 1 )(2 2 ln(2) 7 (3))4 22 ln (2) 1 2 2 2 2 3 l 0 (2 l ) (E. 39 ) 2 423 4 1 3ln(2) 360 16Li 4 (),2 (1 l )() l 2 9(3) (1 ) 1 222 ( 1 ) 1 ln(2) ( 1 ) 49 4 2 22 6 2 3 2 360 l 1 l ( E. 40 ) 1 2 21 4 1 3 ln (2) 3 ln (2) 8Li 4 (2 ), (1 l )() l 2 21(3) (1 ) 1 222 ( 1 ) ln(2) ( 1 ) 49 4 1 2 22 2 2 2 180 l 1 (l 2 ) (E. 41 ) 2 2 22 4 1 3 ln(2) 3 ln(2) 16Li 4 (2 ), and (l 1 ) 2 7(3)()1 222 ln(2) 2 ln(2) 423 4 16Li ().1 (E. 42 ) 1 3 2 3 3 360 4 2 l 0 (l 2 )
Note that ( E. 42 ) is required for the proofs of ( E. 8 ), ( E. 31 ), ( E. 33 ), ( E. 40 ) and ( E. 41 ) . A lso, the separation of ( E. 38 ) into its even and odd components is required for the derivation of ( E. 40 ) and ( E. 41 ).
45 Oct. 1 5 , 2017.
From [ 2 , Eq. (3.71)] and [ 2 , Eq. (3. 93 )] , among many other sums expressed in terms of harmonic numbers , we find two interesting and representative results 2 (l 1) 101 61 4 2 2 22680 36 (3) (E. 43 ) l 1 (l 1) 1 2 (l ) 2 2 (2 8 (3)) (E. 44 ) 2 1 2 l 1 (l 4 ) Also, see ( E. 24 ) and ( E. 9 ) . And finally, from [ 38 ] ( in respective order: Eqs. (3.2a), (2.5c) – incorrect: the negative sign in the summand should be a plus sign – and (2.5d) – incorrect – the right - hand si de should be preceded by a minus sign) we have, making use of ( 66) with n=4 2 (l 1) 21 27 1 4 3 (6 )(3) 2 (5) 36 (E. 45 ) l 1 l
(l 1) 1 2 9 3 3 (3) 2 (5) (E. 46 ) l 1 l and ()l () l 1 27 7 4 2 3(3) 2 (5) 360 . (E. 47 ) l 1 l
46