Research & Reviews Discrete Mathematical Structures vol 3 issue 3

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STM JOURNALS

1. Fractional Calculus of the R-Series
Mohd. Farman Ali, Manoj Sharma 1
2. R-L F Integral and Triple Dirichlet Average of the R-Series
Mohd. Farman Ali 6
3. Dirichlet Average of New Generalized M-series and Fractional Calculus
Manoj Sharma 13
4. A Brief Review on Algorithms for Finding Shortest Path of Knapsack Problem
Swadha Mishra 17
5. Research and Industrial Insight: Discrete Mathematics
20
ContentsResearch & Reviews: Discrete Mathematical Structures

RRDMS (2016) 1-5 © STM Journals 2016. All Rights Reserved Page 1
Volume 3, Issue 3
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Fractional Calculus of the R-Series
Mohd. Farman Ali1,
*, Manoj Sharma2
1
Department of Mathematics, Madhav University, Sirohi, Rajasthan, India
2
Department of Mathematics, Rustamji Institute of Technology, BSF Academy, Tekanpur, Gwalior,
Madhya Pradesh, India
Abstract
The present paper creates a special function called as R-series. This is a special case of H-
function given by Inayat Hussain. The Hypergeometric function, Mainardi function and M-
series follow R-series and these functions have recently found essential applications in solving
problems in physics, biology, bio-science, engineering and applied science etc.
Mathematics Subject Classification—26A33, 33C60, 44A15
Keywords: Fractional calculus operators, R−series, Mellin-Barnes integral, special functions
INTRODUCTION TO THE H-FUNCTION
The H- function of Inayat Hussain, is a generalization of the familiar H-function of Fox, defined in
terms of Mellin-Barnes contour integral [1], as
𝐻 𝑝,𝑞
𝑚,𝑛
[𝑧(𝛽 𝑗, 𝐵 𝑗)1,𝑚 ,(𝛽 𝑗, 𝐵 𝑗; 𝑏 𝑗 ) 𝑚+1,𝑞
(𝛼 𝑗, 𝐴 𝑗;𝛼 𝑗)1,𝑛 ,(𝛼 𝑗, 𝐴 𝑗; 𝛼 𝑗) 𝑛+1,𝑝
] =
1
2𝜋𝑖
∫ 𝜃(𝑠)𝑧 𝑠
𝑑𝑠
+𝑖∞
−𝑖∞
(1)
Where the integrand (or Mellin transform of the H-function)
𝜃(𝑠) =
∏ Γ(𝛽 𝑗− 𝐵 𝑗 𝑠) ∏ [Γ(1=𝛼 𝑗+𝐴 𝑗 𝑠)]
𝛼 𝑗𝑛
𝑗=1
𝑚
𝑗=1
∏ [Γ(1=𝛽𝑗+𝐵 𝑗 𝑠)]
𝑏 𝑗𝑛
𝑗=𝑚+1 ∏ Γ
𝑝
𝑗=𝑛+1 (𝛼 𝑗− 𝐴 𝑗 𝑠)
(2)
Contains fractional powers of some of the involved Γ −functions. Here 𝛼𝑗(1𝑒𝑟𝑒𝑝) and 𝛽𝑗(1𝑛𝑑 𝑞) are
complex parameters; 𝐴𝑗 > 0 ( 10𝑟𝑒𝑝 ), 𝐵𝑗 > 0(1 … … … 𝑞) ; and exponents; 𝑎𝑗(𝑗 = 1 … … … 𝑛) and
𝑏𝑗(𝑗 = 1 … … … 𝑞) can take noninteger values. Evidently, when all the exponent 𝑎𝑗 and 𝑏𝑗 take integer
values only, the H-function reduces to the familiar H-function of Fox, [1–3]. The sufficient conditions
for the absolute convergence of the contour integral (1), as given by Buschman and Srivastava, are as
follows [3]:
Ω = 𝐵𝑗
𝑚
𝑗=1 + 𝑎𝑗 𝐴 𝑗 −𝑛
𝑗=1 𝑏𝑗 𝐵𝑗
𝑝
𝑗=𝑚+1 − 𝐴𝑗
𝑞
𝑗=𝑛+1 > 0 and arg(𝑧) <
1
2
𝜋 Ω
THE R-SERIES
The R- series is
𝑅 𝑞
𝛼,𝛽
𝑝
0
(𝑎1 . . . . 𝑎 𝑝; , 𝑏1 . . . . 𝑏 𝑞; 𝑧0
) = 𝑅 𝑞
𝛼,𝛽
𝑝
0
(𝑧)
𝑅 𝑞
𝛼,𝛽
𝑝
0
(𝑧) =
(𝑎1) 𝑘 . . . . .(𝑎 𝑝)
𝑘
(𝑏1) 𝑘 . . . . .(𝑏 𝑞)
𝑘
∞
𝑘=𝑜
𝑧 𝑘
Γ(𝛼𝑘+𝛽)𝑘!
(3)
Here, 𝑝 upper parameters 𝑎1, 𝑎2, . . . . 𝑎 𝑝 and 𝑞 lower parameters 𝑏1, 𝑏2, . . . . 𝑏 𝑞 , 𝛼𝜖𝐶 , 𝑅(𝛼) >
0, 𝑚 > 0 and (𝑎𝑗) 𝑘 (𝑏𝑗) 𝑘

Volume 3, Issue 3
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R-L F Integral and Triple Dirichlet Average of the
R-Series
Mohd. Farman Ali
Department of Mathematics, Madhav University, Sirohi, Rajasthan, India
Abstract
In this article, we establish the relation between some results of triple Dirichlet average of the
R-series and fractional operators. We use a new special function called as R-series, which is a
special case of H-function given by Inayat Hussain. In this article, the solution is obtained in
compact form of triple Dirichlet average of R-series as well as conversion into single
Dirichlet average of R-series, using fractional integral.
Keywords: Dirichlet averages, special functions, R-series and Riemann-Liouville fractional
integral
Mathematics Subject Classification: 2000: Primary: 33E12, 26A33; Secondary: 33C20,
33C65.
INTRODUCTION
The Dirichlet average of a function is a certain kind integral average with respect to Dirichlet
measure. The concept of Dirichlet average was introduced by Carlson in 1977. Carlson has defined
Dirichlet averages of functions, which represent certain types of integral average with respect to
Dirichlet measure [1–4]. He showed that various important special functions could be derived as
Dirichlet averages for the ordinary simple functions like 𝑥 𝑡
,𝑒 𝑥
etc. He has also pointed out that the
hidden symmetry of all special functions, which provided their various transformations can be
obtained by averaging 𝑥 𝑛
,𝑒 𝑥
etc. [5, 6]. Thus, he established a unique process towards the unification
of special functions by averaging a limited number of ordinary functions [7].
Gupta and Agarwal found that averaging process is not altogether new but directly connected with the
old theory of fractional derivative [8, 9]. Carlson overlooked this connection whereas he has applied
fractional derivative in so many cases during his entire work. Deora and Banerji have found the
double Dirichlet average of ex
by using fractional derivatives and they have also found the triple
Dirichlet average of xt
by using fractional derivatives [10, 11].
Sharma and Jain obtained double Dirichlet average of trigonometry function cos 𝑥 using fractional
derivative and they have also found the triple Dirichlet average of ex
by using fractional calculus
[12–15].
Recently, Kilbas and Kattuveetti established a correlation among Dirichlet averages of the generalized
Mittag-Leffler function with Riemann-Liouville fractional integrals and of the hyper-geometric
functions of many variables [16].
DEFINITIONS AND PRELIMINARIES
Some definitions are necessary in the preparation of this paper.
Standard Simplex in 𝑹 𝒌
, 𝒌 ≥ 𝟏:
The standard simplex in 𝑅 𝑘
, 𝑘 ≥ 1 by [1].
𝐸 = 𝐸 𝑘 = {𝑆(𝑢1, 𝑢2, … 𝑢 𝑘) ∶ 𝑢1 ≥ 0, … 𝑢 𝑘 ≥ 0, 𝑢1 + 𝑢2 + ⋯ + 𝑢 𝑘 ≤ 1}

Volume 3, Issue 3
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Dirichlet Average of New Generalized
M-series and Fractional Calculus
Manoj Sharma
Department of Mathematics, Rustamji Institute of Technology, BSF Academy, Tekanpur,
Gwalior, Madhya Pradesh, India
Abstract
We know that every analytic function can be measured as a Dirichlet average and connected
with fractional calculus. In this note, we set up a relation between Dirichlet average of new
generalized M-series, and fractional derivative. Fractional derivative is a derivative of
arbitrary order i.e. may be real, complex, integer or fractional order.
Mathematics Subject Classification: 26A33, 33A30, 33A25 and 83C99.
Keywords: Dirichlet average new generalized M-series, fractional derivative, fractional
calculus operators
INTRODUCTION
Carlson has defined Dirichlet average of functions which represents certain types of integral average
with respect to Dirichlet measure [1–5]. He showed that various important special functions can be
derived as Dirichlet averages for the ordinary simple functions like𝑥 𝑡
,𝑒 𝑥
etc. He has also pointed out
that the hidden symmetry of all special functions which provided their various transformations can be
obtained by averaging 𝑥 𝑛
,𝑒 𝑥
etc. [6, 7]. Thus he established a unique process towards the unification
of special functions by averaging a limited number of ordinary functions. Almost all known special
functions and their well-known properties have been derived by this process.
In this paper, the Dirichlet average of new generalized M-series has been obtained.
DEFINITIONS
We give below some of the definitions which are necessary in the preparation of this paper:
Standard Simplex in 𝑹 𝒏
, 𝒏 ≥ 𝟏:
We denote the standard simplex in 𝑅 𝑛
, 𝑛 ≥ 1 by Carlson [1].
𝐸 = 𝐸 𝑛 = {𝑆(𝑢1, 𝑢2, … … . . 𝑢 𝑛) ∶ 𝑢1 ≥ 0, … … … . 𝑢 𝑛 ≥ 0, 𝑢1 + 𝑢2 + ⋯ … … + 𝑢 𝑛 ≤ 1} (1)
Dirichlet Measure
Let 𝑏 ∈ 𝐶 𝑘
, 𝑘 ≥ 2 and let 𝐸 = 𝐸 𝑘−1 be the standard simplex in 𝑅 𝑘−1
. The complex measure 𝜇 𝑏 is
defined by 𝐸[1].
𝑑𝜇 𝑏(𝑢) =
1
𝐵(𝑏)
𝑢1
𝑏1−1
… … … … … . 𝑢 𝑘−1
𝑏 𝑘−1−1
(1 − 𝑢1 − ⋯ … … … − 𝑢 𝑘−1)𝑏 𝑘
−1
𝑑𝑢1 … … … … . 𝑑𝑢 𝑘−1 (2)
It will be called a Dirichlet measure.
Here,
𝐵(𝑏) = 𝐵(𝑏1, … … … . 𝑏𝑘) =
Γ(𝑏1) … … … … … . . Γ(𝑏 𝑘)
Γ(𝑏1 + ⋯ … … . . +𝑏 𝑘)
,
𝐶> = {𝑧 ∈ 𝑧: 𝑧 ≠ 0, |𝑝ℎ 𝑧| < 𝜋
2⁄ },
Open right half plane and 𝐶>k is the 𝑘 𝑡ℎ
Cartesian power of 𝐶>.

Volume 3, Issue 3
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A Brief Review on Algorithms for Finding Shortest Path
of Knapsack Problem
Swadha Mishra*
Department of Computer Applications, Invertis University, Bareilly, Uttar Pradesh, India
Abstract
The gathering knapsack and knapsack problems are summed up to briefest way issue in a
class of graphs. An effective calculation is used for finding briefest ways that bend lengths are
non-negative. A more effective calculation is portrayed for the non-cyclic which incorporates
the knapsack issue.
Keywords: knapsack problem, rucksack problem
INTRODUCTION
The knapsack problem or rucksack problem is
an issue in combinatorial enhancement: Given
an arrangement of things, each with a weight
and an esteem, decide the number of
everything to incorporate into an accumulation
so that the aggregate weight is not exactly or
equivalent to a given point of confinement and
the aggregate esteem is as expansive as could
reasonably be expected.
Group knapsack problem has been given to:
Minimise ∑ 𝑐𝑛
𝑗=1 jjxj (1)
Subject to ∑ 𝑐𝑛
𝑗=1 jgj=g0 (2)
Where, x1,…, xn non-negative integers.
g0,…, gn are the subset of the elements of a
finite additive abelian group H and c1,…,cn are
non-negative reals.
This algorithm for solving this problem has
been described by Gomory [1], Shapiro [2, 3],
Hu [4] and others. It can be formulated as a
shortest path problem in the following way.
Let G1 be the graph with node H and arc of the
form (h, h+gj) h an arbitrary element of H and
j=1,…, n. The length of such an arc is cj. Let P
be the from 0 to g0 in G1 then if xj is the
number of arcs of the form (h, h+gj) in P then
(x1,…, xn) is a solution to Eq. (2) and the
length of P is Eq. (1). Conversely, if (xn,…, xn)
satisfied Eq. (2), then one may construct paths
from 0 to g0. Now a new algorithm is given in
this paper to solve this problem [5].
The name knapsack problem applies to:
Maximize ∑ 𝑐𝑛
𝑗=0 jjxj (3)
Subject to ∑ 𝑤𝑛
𝑗=0 jxj=W, (4)
Where, x0, xi,…, xn non-negative integers.
Where, c0=0, c1,…, cn are positive reals, w0=1
and w1,…, wn, W are positive integers.
One can formulate a knapsack problem as a
longest path problem defining the graph G2
with nodes 0, 1,…, W and arcs of the form (w,
w+wj) of length cj. The knapsack problem is
then equivalent to that of finding a longest
path from 0 to W [6].
ALGORITHM
The graph G1 and G2 of the previous section
are examples of a class of graphs which for the
purposes of this paper, we call knapsack
graphs.
Definition
A graph G with nodes N and arcs A is a
knapsack graph if,
(1) The arcs A can be partitioned into n
disjoint sets A1,…,AN;
(2) The length of each arc belonging to Aj is lj;
(3) Let P=(i0, i1,...,ip) be a path between an
arbitrary pair of nodes i0, ip.
Suppose that (it-1, it)∈Amt, for t=1,…., p.
Then for any re-ordering, n1,…., np of the
indices m1,…, mp, there exist a path
Q=(j0,j1,…, jp), where j0=i0, jp=ip and (jt-1=jt)
∈Ant for t=1,…., p.
For shortest path problems with non-negative

Volume 3, Issue 3
www.stmjournals.com
Research and Industrial Insight: Discrete Mathematics
Math’s Maze Runner
Mazes are in vogue right now, from NBO's
West world, to the arrival of the British faction
TV arrangement, The Crystal Maze. Be that as
it may, labyrinths have been around for
centuries and a standout amongst the most
acclaimed labyrinths, the Labyrinth home of
the Minotaur, assumes a featuring part in
Greek mythology.
The most part acknowledged that a maze
contains just a single way, regularly spiraling
around and collapsing back on it, in constantly
diminishing circles, though a labyrinth
contains expanding ways, giving the voyager
decisions and the potential for getting,
exceptionally lost.
Design a maze is very tough task and for that
human should be rewarded. Many algorithms
for maze have been created by computer
scientists and mathematicians. These
algorithms based on two principles: one which
begin with a solitary, limited space and
afterward sub-separate it with dividers (and
entryways) to deliver ever littler sub-spaces;
and others which begin with a world brimming
with detached rooms and after that devastate
dividers to make ways/courses between them.
Escape Plan
There are many techniques by which you can
run away from maze but firstly you need to the
details about the maze from which you want to
escape. Most strategies work for "basic"
labyrinths, that is, ones with no tricky alternate
routes by means of scaffolds or "entry circles"
– round ways that lead back to where they
began.
Along these lines, accepting it is a
straightforward labyrinth, the strategy that
many individuals know is "follow the wall”.
Basically, you put one hand on a mass of the
labyrinth (it doesn't make a difference which
hand the length of you are reliable) and after
that continue strolling, keeping up contact
between your hand and the divider. In the end,
you will get out. This is on account of on the
off chance that you envision getting the mass
of a labyrinth and extending its edge to
evacuate any corners, you will in the long run
frame something circle-like, a portion of
which must shape part of the labyrinth's
external limit. This strategy for escape may
not work, be that as it may, if the begin or
complete areas are in the labyrinth's middle.
Be that as it may, a few labyrinths are
purposely intended to disappoint, for example,
the Escot Gardens' beech fence labyrinth in
Devon, which contains no less than five
extensions, thus a long way from
"straightforward".
There is another method by which maze
escape will be easy i.e., Tremaux’s Algorithm,
it works in all the cases.
Envision that, as Hansel and Gretel in the pixie
story, you can leave a trail of "breadcrumbs"
behind you as you explore your way through
the labyrinth and afterward recall these
guidelines: in the event that you touch base at
an intersection you have not beforehand
experienced (there will be no scraps as of now
on the trail ahead), then haphazardly select an
approach. On the off chance that that leads you
to an intersection where one way is different to
you yet the other is not, then select the
unexplored way. What's more, if picking
between an on more than one occasion utilized
way, pick the way utilized once, then leave
another, second trail behind you. The cardinal
administer is never, ever select a way as of
now containing two trails. This technique is
ensured, in the end, to get you out of any
labyrinth.
Mazes in everyday life
Now you think how is this maze thing useful
for us? Maze is interesting and adventurous
but not for our daily day-to-day life when we
are on work or we are doing something
important.
Bill Hillier, a theorist in 1980s, find that most
of the housing estates that have a layout like
maze. This reise q question that we can
measure the maze-iness of a house?

Research & Reviews Discrete Mathematical Structures vol 3 issue 3

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Research & Reviews Discrete Mathematical Structures vol 3 issue 3