what do you want to hear from me? i can only say,that you have problem with your identity! thats sure ! you will not accepting,that your name is turkisch,and its original turkish name !!!!!
THIS FORM OF WRITING IS ONLY GOVERNORS THAT WROTES YOUR FAMILYS NAME WHEN THEY HAVE IMIGRATES!
dont forget,during the ottoman empire,they used arabic alphabets,and also other turkish kingdoms or republic used arabic alphabet to translate to latin alphabet is problematic for officers of the state to write correct, so wost f the names changed their writing form because it has written how it spelled for !!!! and this is your problem to accepting it!! because your name was written in spelling for of ,how the officer can heard the name from the imigrants !!
you must accept that most of the names was written wrong,to know how it it will be written or hearing is different things! I KNOW THAT YOUR NAME IS TURKISH;SO YOU ARE FROM THE TURKS! if you accept or not,you must make peace with your self an not with me! and dont forget it is also possible that you family must not been lived in turkish lands in the world,when they imigrate to northern america,see for president of the slovenia,Mr. turk is one of the children form the ottomans ,may be his grandfather was soldier during the ottoman empire and was arrested from the christians and sends out of the borders,also in germany ,poland,russia,ukrain,moldavia,and other where lives too many turks but most of them have forgotten their identity,but if you look the familynames,so you can explain it!!!
writing form for you only to understand it:
ai = ay ghin or ghn = GIN (withou point on "i")
so correct writing = KAYGIN
for otherone that cant turkish language spelling is kaygin or kaigin !!!
make peace with your self,may be its too hard for you to accept that you are one turk!!!
see also that in different turkishland,like,TURKEY,aserbaijan,tajikistan,turkmenistan,kirgzistan, usbekistan,afganistan(50% usbek-TURKS),east-turkestan(OCCUPATED FROM CHINA),northern mongolia and other turkc states in russian federations will be written different,they are also turks but writing form is different also there are too many dialekts !!! i hope you cant follow my arguments and understand it,for me to explain you in english is too hard for me,is not my mother-language,if you understand in german or turkish i would explain you much more...
but important is,if you are accepting your self or not! AND IF YOU HAVE UNDERSTAND THE PRINCIPE OF THE WRITING FORM !
i only explain you the REAL truth !!! you are looking for valid sources, but this is valid sources !!!