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In this paper, the author defines the function B¯i,jδ,k(n), the number of singular overpartition pairs of n without multiples of k in which no part is divisible by δ and only parts congruent to ± i, ± j modulo δ may be overlined.

Andrews introduced to combinatorial objects, which he called singular overpartitions and proved that these singular overpartitions depend on two parameters δ and i can be enumerated by the function C¯δ,i(n), which gives the number of overpartitions of n in which no part divisible by δ and parts ≡ ± i(Mod δ) may be overlined.

Using classical spirit of q-series techniques, the author obtains congruences modulo 4 for B¯2,48,3(n), B¯2,48,5 and B¯2,412,3.

The results established in this work are extension to those proved in Andrews’ singular overpatition pairs of n.

Access to finance, especially by the poor and marginalized section of the population, is a prerequisite for creating employment opportunities, economic growth, poverty reduction and social cohesion. Access to finance makes transactions quicker, cheaper and safer. Most people around the world having an account in a formal financial institution serve as an entry point into formal financial sector. This study aims to analyze the status of financial inclusion in Assam with respect to demographic penetration, geographic penetration and usage ratio, i.e. credit–deposit ratio.

The study covers a period of 12 years from 2007–08 to 2018–19. Both the parametric and non-parametric statistical tools have been used to analyze the various dimensions of financial inclusion.

The study clearly indicates that there is a significant difference between Assam and aggregate India in financial inclusion and the status of Assam is somewhat lower as compared to the aggregate financial inclusion status of India. To achieve a satisfactory level of financial inclusion, it is not enough to open a bank account for the excluded people, but banks must look at flexibility and timeliness in services to offer a complete package to this segment of the population.

The study is a significant attempt to meet the shortcomings and improve banking coverage for achieving financial inclusion.

Both branch and automated teller machine (ATM) are playing a crucial role in banking coverage expansion in India. People prefer to go to an ATM for withdrawal of money rather waiting in a queue for hours at a branch. Without the existence of a full-fledged brick-and-mortar branch, ATM also plays an important role by providing basic banking services. In India, a significant part of the population is excluded from banking access. The present study aims to investigate how the branch and ATM penetration influence financial inclusion.

The study covers the period from 2008–2009 to 2019–2020. With the application of Welch's t-test, a comparative study is being conducted between branch and ATM. Further, with the application of regression analysis, the study analyses how the branch and ATM network expansion influence financial inclusion.

Though in recent times customers prefers to visit an ATM and its growth rate is higher than branches, the study found no significant differences between the growth of branch and ATM. Further, results of regression show both branches and ATMs have significant impacts on financial inclusion.

In micro concept both have a common role in respect of service provided to customers. While in macro concept a list of specific services can be provided through branch level only. This study has a significant role, considering the importance of branches or ATMs and cost of installing a physical branch.

An inclusive financial system is essential to develop the country’s economy. A massive shift in financial inclusion was observed by the initiative of government to include financially excluded into the formal financial system by launching Pradhan Mantri Jan Dhan Yojana (PMJDY) in 2014. This paper aims to attempt to examine the efficiency of public sector banks in financial inclusion during pre and post introduction of PMJDY.

The data envelopment analysis is used to measure the efficiency of the banks towards financial inclusion for the periods, 2010–2011 to 2013–2014 as pre-introduction and 2014–2015 to 2017–2018 as post-introduction phase. For this study, supply-side parameters of financial inclusion considered as input variables and demand-side parameters as output variables.

The study finds that overall average efficiency towards financial inclusion increases significantly during post-phase, though all the public sector banks are not performing equally. There is a significant variation in efficiency level between them and even between the two periods. Further, there is a huge opportunity to enhance technical efficiency with the same quantity of input which will help to achieve the target of financial inclusion.

A comparative study between the two phases has taken place to analyse the impact of the scheme on the technical efficiency of banks. One of the notable innovativeness of this study is that, unlike most of the previous studies which are mostly theoretical and conceptual, the present study may place itself as a unique inquiry in the domain of efficiency review of public sector banks during pre and post introduction of PMJDY.